Quantitative Techniques in Businessjnujprdistance.com/assets/lms/LMS JNU/BBA/Sem III...With the help of matrices & determinants, we can solve equations & system of equations. 1.2 Matrix

Quantitative Techniques in Business

This book is a part of the course by Jaipur National University, Jaipur.This book contains the course content for Quantitative Techniques in Business.

JNU, JaipurFirst Edition 2013

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Index

ContentI. ...................................................................... II

List of TablesII. .......................................................... VII

Case StudyIII. .............................................................. 76

Solved ExamplesIV. ................................................... 79

BibliographyV. ............................................................ 86

Self Assessment AnswersVI. ....................................... 89

Book at a Glance

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Content

Chapter I ........................................................................................................................................................IMatrices & Determinants .............................................................................................................................1Aim .................................................................................................................................................................1Objective .........................................................................................................................................................1Learning Outcome ......................................................................................................................................... 11.1 Introduction .............................................................................................................................................. 21.2 Matrix ....................................................................................................................................................... 21.3 Types of Matrix ........................................................................................................................................ 2 1.3.1 Row Matrix .............................................................................................................................. 2 1.3.2 Column Matrix ......................................................................................................................... 2 1.3.3 Square Matrix .......................................................................................................................... 2 1.3.4 Zero or Null Matrix .................................................................................................................. 2 1.3.5 Diagonal Matrix ....................................................................................................................... 3 1.3.6 Unit or Identity Matrix ............................................................................................................. 31.4 Operations on Matrices ............................................................................................................................ 3 1.4.1 Addition of Two Matrices ........................................................................................................ 3 1.4.2 Subtraction of Two Matrices .................................................................................................... 3 1.4.3 Scalar Multiplication of a Matrix ............................................................................................. 3 1.4.4 Multiplication of Two Matrices ............................................................................................... 41.5 Determinants ............................................................................................................................................ 4 1.5.1 Singular and Non-singular Matrices ........................................................................................ 41.6 Properties of Determinants ...................................................................................................................... 41.7 Difference between Matrices & Determinants......................................................................................... 51.8 Uses & Applications of Matrices & Determinants .................................................................................. 51.9 Solutions to System of Linear Equations ................................................................................................. 5Summary ....................................................................................................................................................... 7References ..................................................................................................................................................... 7Recommended Readings ............................................................................................................................. 8Self Assessment ............................................................................................................................................. 9

Chapter II .................................................................................................................................................... IIMathematical Logic ....................................................................................................................................11Aim ...............................................................................................................................................................11Objectives .................................................................................................................................................... 11Learning Outcome ........................................................................................................................................112.1 Introduction ............................................................................................................................................ 12 2.1.1 Statement ............................................................................................................................... 12 2.1.2 Compound statement ............................................................................................................. 12 2.1.3 Truth Table ............................................................................................................................. 122.2 Logical Connectives ............................................................................................................................... 13 2.2.1 Conjunctions .......................................................................................................................... 13 2.2.2 Disjunction ............................................................................................................................. 13 2.2.3 Negation ................................................................................................................................. 13 2.2.4 Conditional ............................................................................................................................. 142.3 Tautology and Contradiction .................................................................................................................. 142.4 Laws of Algebra of Propositions ............................................................................................................ 14Summary ......................................................................................................................................................16References ................................................................................................................................................... 16Recommended Readings ........................................................................................................................... 16Self Assessment ........................................................................................................................................... 17

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Chapter III .................................................................................................................................................. 19Set Theory ................................................................................................................................................... 19Aim .............................................................................................................................................................. 19Objective .......................................................................................................................................................19Learning Outcome ....................................................................................................................................... 193.1 Definition ............................................................................................................................................. 203.2 Standard Sets .......................................................................................................................................... 203.3 Types of Sets .......................................................................................................................................... 20 3.3.1 Finite set ................................................................................................................................. 20 3.3.2 Infinite set: If the set contains an infinite number of elements it is called an infinite set. ..... 20 3.3.3 Null set or Empty set: ........................................................................................................... 20 3.3.4 Universal set: ........................................................................................................................ 203.4 List of Symbols ...................................................................................................................................... 20 3.4.1 Belongs to (∈) ........................................................................................................................ 20 3.4.2 Not Belongs to (∉ ) : x ∉ A : x is not an element of set A..................................................... 20 3.4.3 Equal to ( = ) : A = B : A and B are equal sets. ...................................................................... 20 3.4.4 Not Equal to ( ≠ ) : A≠B : A and B are unequal sets. ............................................................. 20 3.4.5 Subset (⊂) : A⊂ B : Set A is a subset of set B ....................................................................... 21 3.4.6 Not a Subset (⊄) ................................................................................................................... 213.5 Operations on Sets ................................................................................................................................. 21 3.5.1 Intersection (∩) ..................................................................................................................... 21 3.5.2 Complementary Set ( ′ or ) .................................................................................................... 213.6 De Morgan’s Law................................................................................................................................... 223.7 Application ............................................................................................................................................. 22Summary ..................................................................................................................................................... 23References ................................................................................................................................................... 23Recommended Readings ........................................................................................................................... 23Self Assessment ........................................................................................................................................... 24

Chapter IV ................................................................................................................................................. 26Arithmetic Progression & Geometric Progression ................................................................................. 26Aim .............................................................................................................................................................. 26Objective ...................................................................................................................................................... 26Learning Outcome ....................................................................................................................................... 264.1 Introduction: ........................................................................................................................................... 274.2 Arithmetic Progression: ......................................................................................................................... 274.3 Geometric Progression: .......................................................................................................................... 274.4 Sequences ............................................................................................................................................. 27 4.4.1 Definition ............................................................................................................................... 27 4.4.2 nth Term of the Sequence ( t ) n ................................................................................................ 28 4.4.3 Sum of the First n Terms (S ) n ............................................................................................... 284.5 Arithmetic Sequence or Arithmetic Progression .................................................................................... 29 4.5.1 Definition ............................................................................................................................... 29 4.5.2 nth Term of an A.P. .................................................................................................................. 29 4.5.3 Sum of the First n Terms of an A.P. (S ) n .............................................................................. 29 4.5.4 Properties of an A.P. ............................................................................................................... 294.6 Geometric Progression ........................................................................................................................... 304.7 nth Term of a G.P. .................................................................................................................................. 304.8 Sum of the First n Terms of a G.P. (S ) n ................................................................................................ 30Summary ..................................................................................................................................................... 31References ................................................................................................................................................... 32Recommended Readings ........................................................................................................................... 32Self Assessment ........................................................................................................................................... 33

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Chapter V .................................................................................................................................................... 35Probability .................................................................................................................................................. 35Aim .............................................................................................................................................................. 35Objective ...................................................................................................................................................... 35Learning Outcome ....................................................................................................................................... 355.1 Introduction: ........................................................................................................................................... 365.2 Definition ............................................................................................................................................. 365.3 Sample Space & Events: ........................................................................................................................ 365.4 The Venn Diagram: ................................................................................................................................ 365.5 Rules of Probability: .............................................................................................................................. 365.6 Applications: .......................................................................................................................................... 375.7 Conditional Probability: ......................................................................................................................... 37 5.7.1 Independent & Dependent Events: ........................................................................................ 37 5.7.2 Multiplication Rule: ............................................................................................................... 37 5.7.3 Conditional Probability: Independent events ......................................................................... 37 5.7.4 Conditional Probability: Dependent Events .......................................................................... 37 5.7.5 Multiplication Rule: Dependent Events ................................................................................. 375.8 How to calculate the Probabilities? ........................................................................................................ 385.9 Steps to solve probability ....................................................................................................................... 38Summary ..................................................................................................................................................... 40References ................................................................................................................................................... 40Recommended Readings ........................................................................................................................... 40Self Assessment Questions ......................................................................................................................... 41

Chapter VI ................................................................................................................................................. 43Permutation & Combination .................................................................................................................... 43Aim .............................................................................................................................................................. 43Objective ...................................................................................................................................................... 43Learning Outcome ....................................................................................................................................... 436.1 Introduction to principles of counting: .................................................................................................. 446.2 Definition of Permutation: .................................................................................................................... 446.3 Combination: ......................................................................................................................................... 446.4 Fundamental principles of counting: ..................................................................................................... 44 6.4.1 Addition Rule : ....................................................................................................................... 44Summary ..................................................................................................................................................... 46References ................................................................................................................................................... 46Recommended Readings ........................................................................................................................... 46Self Assessment ........................................................................................................................................... 47

Chapter VII ................................................................................................................................................ 49Interpolation ............................................................................................................................................... 49Aim .............................................................................................................................................................. 49Objective ...................................................................................................................................................... 49Learning Outcome ....................................................................................................................................... 497.1 Introduction: ........................................................................................................................................... 507.2 Definition of Interpolation: .................................................................................................................... 507.3 Application: ............................................................................................................................................ 507.4 Need & Importance of Interpolation ...................................................................................................... 507.5 Methods of Interpolation: ...................................................................................................................... 51 7.5.1 Graphical Method: ................................................................................................................ 51 7.5.2 Newton’s method of advancing differences: .......................................................................... 51 7.5.3 Lagrange’s Method: ............................................................................................................... 51 7.5.4 Newton-Gauss Foreword Method: ......................................................................................... 51 7.5.5 Newton-Guass Backward Method: ........................................................................................ 51

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Summary ..................................................................................................................................................... 53References ................................................................................................................................................... 53Recommended Readings ........................................................................................................................... 53Self Assessment ........................................................................................................................................... 54

Chapter VIII ............................................................................................................................................... 56Commercial Arithmetic ............................................................................................................................. 56Aim .............................................................................................................................................................. 56Objective ...................................................................................................................................................... 56Learning Outcome ....................................................................................................................................... 568.1 Introduction: ........................................................................................................................................... 578.2 Commission & Brokerage: .................................................................................................................... 57 8.2.1 Application of Commission: .................................................................................................. 578.3 Profit & Loss: ......................................................................................................................................... 578.4 Formulae ............................................................................................................................................. 588.5 Interest ............................................................................................................................................. 58 8.5.1 Interest: .................................................................................................................................. 58 8.5.2 Simple Interest ....................................................................................................................... 58 8.5.3 Compound Interest ................................................................................................................. 588.6 Present Worth ......................................................................................................................................... 588.7 True Discount (T.D.) .............................................................................................................................. 588.8 Sum Due (S.D.) ...................................................................................................................................... 588.9 Insurance ............................................................................................................................................. 59Summary ..................................................................................................................................................... 60References ................................................................................................................................................... 60Recommended Readings ........................................................................................................................... 60Self Assessment ........................................................................................................................................... 61

Chapter IX .................................................................................................................................................. 63Relations & Functions ............................................................................................................................... 63Aim .............................................................................................................................................................. 63Objective ...................................................................................................................................................... 63Learning Outcome ....................................................................................................................................... 639.1 Relation .................................................................................................................................................. 649.2 Domain & Range of a Relation: ............................................................................................................. 649.3 Functions ............................................................................................................................................. 64 9.3.1 Range, image, co-domain ...................................................................................................... 659.4 Break Even Analysis: ............................................................................................................................. 65Summary ..................................................................................................................................................... 66References ................................................................................................................................................... 66Recommended Readings ........................................................................................................................... 66Self Assessment ........................................................................................................................................... 67

Chapter X ................................................................................................................................................... 69Statistics ...................................................................................................................................................... 69Aim .............................................................................................................................................................. 69Objectives .................................................................................................................................................... 69Learning Outcome ....................................................................................................................................... 6910.1 Introduction ............................................................................................................................ 7010.2 Definition of Statistics.......................................................................................................................... 7010.3 Scope and Applications of Statistics .................................................................................................... 7010.4 Characteristics of Statistics .................................................................................................................. 7010.5 Functions of Statistics .......................................................................................................................... 7110.6 Limitations of Statistics ....................................................................................................................... 7110.7 Classification ........................................................................................................................................ 71

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10.8 Objectives of Classification ................................................................................................................. 7110.9 Characteristics of Classification ........................................................................................................... 7210.10 Frequency Distribution ...................................................................................................................... 72 10.10.1 Discrete or Ungrouped Frequency Distribution ................................................................. 72 10.10.2 Continuous or Grouped Frequency Distribution ............................................................... 72 10.10.3 Cumulative Frequency Distribution ................................................................................... 72Summary ..................................................................................................................................................... 73References ................................................................................................................................................... 73Recommended Readings ........................................................................................................................... 73Self Assessments ......................................................................................................................................... 74

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List of Tables

Table 1.1 Difference between Matrices & Determinants ....................................................... 5Table 2.1 Conjunction Table ........................................................................................................................ 13Table 2.2 Disjunction Table ......................................................................................................................... 13Table 2.3 Negation Table ............................................................................................................................. 13

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Chapter I

Matrices & Determinants

AimAim of this chapter is to introduce the concept of matrices & determinants

Objective

Objective of this chapter is

To understand the different types of matrices & determinants•

To bring forward an idea of applications of matrices in various fields•

Learning Outcome

After end of this chapter students will conversant with

What matrices are•

Performing basic operations on matrices•

Special forms of matrices•

The matrix determinant•

How to calculate the different matrices i.e. 2x2, 3x3, 2x3 matrix•


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1.1 IntroductionThe study of matrices & determinants is of immense significance in business & economics. We find the application of matrices & determinants in various topics of economics & management such as linear programming, theory of games, general equilibrium analysis, matrix multiplier, input-output analysis & so.With the help of matrices & determinants, we can solve equations & system of equations.

1.2 MatrixA matrix is simply a set of numbers arranged in a rectangular table.It is written in either ( ) or [ ] brackets. A set of numbers arranged in a rectangular array of ‘m’ rows and ‘n’ columns, of an order m x n (Read as m by n).

A =

Matrices are used to solve problems in:electronics• statics• robotics• linear programming• optimization• intersections of planes• genetics•

Matrix Notation:A matrix is written with ( ) or [ ] brackets. Do not confuse a matrix with a determinant which uses vertical bars | |. A matrix is a pattern of numbers; a determinant gives us a single number. The size of a matrix is written: rows × columns.

1.3 Types of Matrix1.3.1 Row MatrixA matrix having a single row is called a row matrix.A=

1.3.2 Column MatrixA matrix having a single column is called a column matrix.

A=

1.3.3 Square MatrixA matrix having equal number of rows and columns is called a square matrix.

A =

1.3.4 Zero or Null MatrixA matrix having each and every element as a zero is called zero or null matrix.

A =

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1.3.5 Diagonal MatrixA square matrix having all elements zero except the principal diagonal elements is called a diagonal matrix.Matrix elements as a , a , a etc are called principal diagonal elements.

A =

1.3.6 Unit or Identity MatrixA square matrix which is a diagonal matrix having all principal diagonal elements as one (unity).

A=

1.3.7 Equal MatricesTwo matrices A and B are said to be equal if A and B have the same order and each and every element of matrix A is equal to the corresponding element of matrix B.

A=

1.4 Operations on Matrices

1.4.1 Addition of Two MatricesNecessary Condition: The two matrices must be of same order.PropertiesConsider A, B, C matrices having the same order.• (A+B) =( B+A)• A+(B+C) = (A+B)+C• A+ zero matrix = A

1.4.2 Subtraction of Two MatricesNecessary ConditionThe two matrices must have the same order.

PropertiesConsider A, B and C are the matrices having same order.

A-B = -(B-A)• A-(B-C) = (A-B)-C=(A-C)-B• A - zero matrix = A• A A = zero matrix•

1.4.3 Scalar Multiplication of a MatrixNecessary Condition: Scalar must not be zero.

A=


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ProcedureLet any non-zero scalar be K . and matrix A then scalar multiplication is denoted by

KA=

1.4.4 Multiplication of Two MatricesNecessary Condition:

Order is important in matrix multiplication. AB is not always equal to BA.• The number of columns of the first matrix must match the number of rows of the second matrix.• If the order of matrix A is m n and of matrix B is n l then the order of the resultant matrix of AB must be m l•

PropertiesConsider A, B, C are matrices.

AB = BA• ABC = (AB)C = A(BC)• A(B+C) = AB + AC• AI = IA = A•

1.5 Determinants:A determinant of a matrix represents a single number. We obtain this value by multiplying and adding its elements in a special way. We can use the determinant of a matrix to solve a system of simultaneous equations.

For example, if we have the (square) 2 × 2 matrix: then The determinant of this matrix is written within vertical lines as follows:

A=

1.5.1 Singular and Non-singular MatricesA square matrix A is said to be singular if |A| = 0.A square matrix A is said to be non-singular if |A| ¹ 0.Examples1. Consider matrix A = 3 -6 then |A| = 3 -6 = 01 -2 1 -2Therefore, matrix A is called a singular matrix.2. Consider matrix A = 2 4 then |A| = 2 4 = 08-5 -6 -5 -6Therefore, matrix A is called a non-singular matrix.

1.6 Properties of Determinants:The value of determinants remains unchanged if its rows & columns are interchanged• If any two rows/columns of a determinant changes by minus sign only• If any two rows or columns of a determinant are identical, then its value is zero• If each element of a row/column of a determinant is expressed as a sum of two or more terms, then the determinant • can be expressed as the sum of two or more determinants

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If each element of a row/column of a determinant is multiplied by the same constant & then added to the • corresponding elements of some other row/column, then value of determinant remains sameIf each element of a row/column of a determinant is zero, then its value is zero•

1.7 Difference between Matrices & Determinants

Features Matrices Determinants

Definition: A • matrix is simply a set of numbers arranged in a rectangular table.

A • determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of products.

Way of writing It is written in Brackets• It is written in two straight lines•

Result These are resulted in set of numbers • grouped in bracket. These are resulted in single number.•

Affection Scalar multiplication affect all the • numbers in the matrix

Scalar multiplication only affects single • row & single column.

Nature Matrices contain many elements• It has single number.•

Value Matrices can be negative• Determinants are always positive.•

Table 1.1 Difference between matrices & determinants

1.8 Uses & Applications of Matrices & Determinants:Graph theory:•

The adjacency matrix of a finite graph is a basic notion of graph theory. �

Linear combinations of quantum states in physics:• The first model of quantum mechanics by Heesenberg in 1925 represented the theory’s operations by �infinite dimentional matrices acting on quantum states. This is also referred to as Matrix Mechanics.

Computer graphics• 4x4 transformation rotaion matrices are commonly used in computer graphics. �

Solving linear equations:• Using row reduction �Cramer’s rule (Determinants) �Using the inverse matrix �

Cryptography:• It consists of Encryption & Decryption. �In Encryption data is converted into some unreadable form. �In decryption data, encrypted data converted into readable form. �


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1.9 Solutions to System of Linear EquationsConsider system of equations as,a x + b y = c ……………..(1) a x + b y = c ……………..(2)

Note:x, y are variables of the equations. a1 ,b1 , a2 , b2 are coefficients of variables c1 , c2 are constants. then the solution can be obtained by Crammer’s Rule as follows:step 1: Solve the determinant of coefficients of variables say delta

=

Step 2: Solve the determinant replacing constants in the place of coefficients of variable x say delta x ( ) x

=

Step 3: Solve determinant replacing constants in the place of coefficients of variable y say delta y ( ) y

=

Step 4: Obtain solution as x = / and y =

Note: For a system of three variables obtain solution as, x = / and y = z= /

Solved Examples:1) Solve 2x+3y = 9-x + y = -2Using Crammer’s Rule

=

= 2+3 = 5

=

= 9+6 = 15 x

=

= -4+9=5

Therefore, solution is as x = / = 15/5 = 3 x y = = 5/5 = 1 y

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SummaryA matrix is defined as a rectangular array of elements.•

If the arrangement has m rows and n columns, then the matrix is of order mxn (read as m by n). �A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter. �

Two matrices are said to be comparable if they have the same order.• Addition and subtraction of two matrices is possible only if they have the same order.•

If two matrices A and B are of same order, then A - B = A + (- B). �

Commutative law, associative law holds good for addition of matrices.• The additive identity of a matrix A of order mxn is the zero matrix of order mxn.• The additive inverse of a matrix A is -A.• The multiplication of two matrices A and B is possible if the number of columns of A is equal to the number • of rows B.

Suppose A is a matrix of order mxn and B is a matrix of order nxp, the matrix AB is of order mxp. �

If A, B and C are the matrices which can be multiplied then• Matrix multiplication is not commutative, i.e., AB • BA (always)Associative law holds good for matrix multiplication, i.e., (AB)C = A(BC)• Matrix multiplication is distributive with respect to addition A(B + C) = AB + AC or (A + B)C = AC + BC• If A is a matrix of order mxn and is a scalar (real or complex) then the matrix kA is obtained by multiplying • each element of A by k.to every square matrix, a value can be associated which is known as the determinant of the matrix.Note that the determinant of kA where k is a scalar and A is a square matrix, is given by k• n times determinant of A.i.e., is |kA| = kn |A|The value of the determinant remain unchanged if its rows and columns are interchanged•

If two rows or columns of a determinant are interchanged, then the sign of the determinant is changed. �If any two rows or columns of a determinant are equal, then its value is zero. �If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k. �If two rows or columns of determinant are proportional, the value of the determinant is zero. �A square A = [a � ij] is said to be symmetric if AT = A, i.e., if a � ij = aji

A square matrix A is said skew symmetric if A � T = - A, i.e., aij = - aji

Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix �as follows

For a 2 x 2 matrix, the adjoint is got by interchanging elements in the leading diagonal and changing �signs in the other diagonal.


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ReferencesDr. V.N. Kala, Rajashri Rana. Matrices. 2009.Laxmi Publication ltd. First Edition. P1-25 & 140-144.• TR Jain, SC Agarwal. Business Mathematics & Statistics. V.K Enterprises.2009-10. Revised. P1-40.• TR Jain, SC Aggarwal, Quantitative Methods. 2008-09. FK Publication. P1-92.• J.O.Bird(2001), Newnes engineering mathematics pocket book, Matrices & determinants, Butterworth & • hannmen, p230-240.Gareth Williams (2009), Gareth Williams, Matrices & determinants, Jones& Bartlett Publications, 7• th Edition, p69-165. David Poole(2005), Linear Algebra, Matrix, Cengage learning, 2• nd Edition, p134-150.

Recommended ReadingsDavid McMahon, Linear Algebra Demystified, Matrices, McGraw-hill publication, 2005, p34-74• Howard Anton, Elementary Linear Algebra, Matrices, FM Publications, 10• th edition 2010, p1-106.Warner Greub, Linear Algebra graduate texts in mathematics, Springer, 1975, p83-131 •

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Self AssessmentGiven the Matrix P= 1. & aij is the element of matrix P in the ith row & the jth column, State the value of a21.

2 a. 3 b. 4c. 5d.

It is given that P=2. and Q= . Find the value of x+y, if P=Q.3a. 5b. 6c. 8d.

A _______ is rectangular array of numbers/elements enclosed in large brackets.3.

Square Matrix a. Determinantb. Paranthesisc. Matrixd.

Addition or Subtraction of two matrices can only be carried out if they are of the _______.4. different ordera. same orderb. complimentaryc. greater or smallerd.

The multiplication of matrix by a number is called __________.5. rectangular multiplicationa. scalar multiplicationb. square multiplicationc. multiplicationd.

Choose correct option from the following.6. The matrix A=

Symmetrica. Square matrixb. Null matrixc. Diagonal Matrixd.

Using determinants, find the area of triangle whose vertices are7. (2, -7), (1, 3), (10, 8).

-47.5a. 47.5b. 45.0c. 45.9d.


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Add the following two matrices8. A = B =

a.

b.

c.

d.

Multiply the matrix9.

a.

b.

c.

d.

Find the inverse of A10. =

a.

b.

c.

d.

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Chapter II

Mathematical Logic

AimAim of this chapter is to introduce the mathematical logic in management perspective by defining the areas of • foundation of mathematics

Objectives


To treat the operations of formal logic in symbolical & algebraic way•

To understand the logical connectives to describe the statements•

To explain different laws of algebra of prepositions•

To enrich the different statement patterns in mathematical logic•

Learning Outcome

At the end of this chapter students will able to

Use the mathematical logic in defining management problems•

Solve the complex procedures into simpler form•

Apply the mathematical logic & its applications to analyze & define concepts of management•


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2.1 IntroductionMathematical logicMathematical Logic is a tool for providing precise meaning to mathematical statements. It includes:

A formal language for expressing them.• A concise notation for writing them.• A methodology for objectively reasoning about their truth or falsity•

2.1.1 Statement

A • statement/simple statement, or a proposition, is a declarative sentence that is either true or false, but not bothSimple statement is basic building block of logic•

2.1.2 Compound statement

• Compound statement is combination of two or more statements. E.g. Today is Friday and Today is Holiday

2.1.3 Truth Table

The value of statement is represented by Truth table. Only TRUE (T)• & FALSE (F) is appeared in Truth Table.The compound statements are basically connected with the connectives:•

Negation �Conjunction �Disjunction �Conditional �Bi-conditional �

Symbols • p, q, r, called statement variables Symbols ~, • ∧, ∨, →, and ↔ are called logical connectivesIf • A and B are statement formulas, then the expressions (~A), (A ∧ B), (A ∨ B), (A → B) and (A ↔ B) are statement formulasExpressions are statement formulas that are constructed only by using above statements•

Truth value (Truth)One of the values “truth” or “falsity” assigned to a statement•

True is abbreviated to T or 1 �False is abbreviated to F or 0 �

Negation (Falsity)• The negation of p, written � ∼p, is the statement obtained by negating statement p Truth values of p and � ∼p are oppositeSymbol ~ is called “not” ~p is read as “not p” �

Example: p: A is a consonant.~p: it is the case that A is not a consonant.

Precedence of logical connectives is:~ highest∧ second highest

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∨ third highest → fourth highest↔ fifth highest

2.2 Logical Connectives:2.2.1 Conjunctions:

Conjunction is the combination of statements using “AND”• The conjunctions of two statements are TRUE, only if each component is True• It is represented by the sign ‘^’•

p q p^qT T TT F FF T FF F F

Table 2.1 Conjunction table

2.2.2 Disjunction:

Disjunction is the combination of statements using “OR”• The conjunction of two statements is true if either one component is true• It is represented as sign ‘v’•

p q pvqT T T

T F T

F T T

F F F

Table 2.2 Disjunction table

2.2.3 Negation:

Negation is the NOT of a simple statement• The Truth value of negation of a statement is the opposite of the truth value of the original statement• It is represented as sign ‘~’•

p -pT FF T

Table 2.3 Negation table


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2.2.4 Conditional:

Conditional statement is the statement in the form, if p, then “p implies q”• The conditional p• q is true unless p is true & q is FalseIt is represented as sign “• ”

p q p qT T TT F FF T TF F T

Table 2.4 Conditional table

Bi-conditional:

Bi-conditional is statement in the form p if and only if q or p if q• If p & q have the same value, p↔q is true, otherwise will be False• It is represented as sign “↔ •

p q p↔ qT T TT F F

F T F

F F T

Table 2.5 Bi-conditional able

2.3 Tautology and ContradictionTautology: A statement is said to be a tautology if it is true for all the truth value of its components• Contradiction or Fallacy: A statement is said to be a contradiction if it is false for All the truth value of its • components

2.4 Laws of Algebra of PropositionsIdentity:p V p ≡ p p Λ p ≡ p p → p ≡ T p ↔ p ≡ T p V T ≡ T p Λ T ≡ p p → T ≡ T p ↔ T ≡ p p V F ≡ p p Λ F ≡ F p → F ≡ ~p p ↔ F ≡ ~p T → p ≡ p F → p ≡ T

Commutative:p V q ≡ q V p p Λ q ≡ q Λ p p → q ≠ q → p p ↔ q ≡ q ↔ p Complement:p V ~p ≡ T p Λ ~p ≡ F p → ~p ≡ ~p p ↔ ~p ≡ F ~p → p ≡ p

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Double Negation:~(~p) ≡ p

Associative:p V (q V r) ≡ (p V q) V r p Λ (q Λ r) ≡ (p Λ q) Λ r

Distributive:p V (q Λ r) ≡ (p V q) Λ (p V r) p Λ (q V r) ≡ (p Λ q) V (p Λ r)

Absorbtion:p V (p Λ q) ≡ p p Λ (p V q) ≡ p De Morgan’s law:~(p V q) ≡ ~p Λ ~q ~(p Λ q) ≡ ~p V ~q Equivalence of Contrapositive:p → q ≡ ~q → ~p

Others:p → q ≡ ~p V q p ↔ q ≡ (p → q) Λ (q → p)


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SummaryMathematical logic is logic relevant to the study of mathematics, logic relies on the following:•

A statement (or a proposition) may be described as an assertive sentence, which is either true or false, �but not bothThe statements formed by combining two or more simple statements with logical connectives are called �compound or composite statements. ‘True' or ‘False' are called the truth values of a statement. If a statement is true, then its truth value is ‘True' denoted by ‘T'. If a statement is false, then its truth value is ‘False' denoted by ‘F'It is a tabular form showing the truth values of various simple and compound statements in a number �of rows and columnsThe statements are described in Negation, Conjunction, Implication, Dissjunction.etc �Two statement patterns s and s are said to be logically equivalent if they have identical truth tables �

Tautology: A statement is said to be a tautology if it is true for all the truth value of its components•

ReferencesJoseph S. Fulda (1993). Exclusive Disjunction and the Bi-conditional: An Even-Odd Relationship. Mathematics • Magazine 66 (2):124Philip P. Hallie (1954). A Note on Logical Connectives. Mind 63 (250):242-245• Dean P. McCullough (1971). Logical Connectives for Intuitionist Propositional Logic. Journal of Symbolic • Logic 36 (1):15-20.Heinrich Wansing (2006). Logical Connectives for Constructive Modal Logic. Synthese 150 (3)•

Recommended ReadingsLeidn university, • http://www.math.leidenuniv.nl/~redig/lecturenotesstatistics.pdf, last accessed date: 14th oct 2010Star, Statistics • http://www.stat-help.com/intro.pdf , last accessed date: 14th oct 2010Richald, Statistics, • http://people.richlan,.edu/james/lecture/m170, last accessed date: 14th oct 2010Art of problem solving, Statistics, • http://www.artofproblemsolving.com/LaTeX/Examples/statistics_firstfive.pdf last accessed date: 14th oct 2010Answers.com, Statistics, • http://www.answers.com/topic/statistics last accessed date: 14th oct 2010

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Self AssessmentJohn likes music & Films.1.

p ^ qa. p b. qp ~ qc. p v qd.

Peter is a doctor or an Engineer.2. p ^ qa. p ~ qb. p c. qp v qd.

John likes Yamaha and Alex likes Hero Honda.3. p a. qp ~ qb. p ^ qc. p v qd.

2+4=6 and 6 is composite number.4. p ~ qa. p ^ qb. p c. qp v qd.

2+6 ≠ 7 or 7 is prime number.5. p ~ qa. p v qb. p c. qp ^ qd.

If a number is not real then it is complex.6. pa. q~b. p ~qpc. ~q d. ~p q

7. If Alex is intelligent or hardworking then logic is easy.(~pa. ~q)(~rb. ~s)(pc. q) rp ^ q = rd.


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If 3 is not odd and 2 is not even then 7 is not odd or 8 is not even.8. (~pa. ~q) (~r ~s)~p v q = ~r v ~sb. p V q = r V sc. (pd. q) r.

Write TRUTH TABLE for following statement9. ~p v ~q

Write TRUTH TABLE for following statement10. p^(~p v ~q)

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Chapter-III

Set Theory

AimAim of this chapter is to understand the concept of Set theory•

Objective


To understand the types of sets•

To explain general framework of set theory•

To understand the properties of set theory•

To emphasize on operations of set theory•

Learning Outcome

At the end of this chapter students will be conversant with following

Elementary operations of sets i.e. Union, Intersection etc•

Formal logic behind set theory is clearly understood•

Elementary fact of set theories helps to explain the functional & binary relation between two sets•

The importance of set theory in applied mathematics such as, Descriptive set theory, Fuzzy set theory, Inner •

model theory, Large Cardinals, Determinacy, Forcing etc


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3.1DefinitionA set is a collection of well-defined objects enclosed in curly brackets, generally denoted by capital letters.The objects which form the set are called elements or members of the set.

e.g. A = {1,2,3,……100}= set of numbers from 1 to 100V = {a,e,i,o,u }= set of all vowels

3.2 Standard SetsN = set of all natural numbers = {1,2,3,4……}• Z = set of all integers = {….. 3,-2,-1,0,1,2,3,…….}• Q = set of all rational numbers = {p/q : p, are integers , q• ≠0}R = set of all real numbers•

3.3 Types of Sets3.3.1Finiteset:Ifthesetcontainsafinitenumberofelementsitiscalledafiniteset.A={10,20,40}3.3.2Infiniteset:Ifthesetcontainsaninfinitenumberofelementsitiscalledaninfiniteset. A= {10,20,40,…………..}

3.3.3 Null set or Empty set: The set having no element is called a null set or an empty set. It is denoted by the symbol ‘φ'.A= or A= {}

3.3.4 Universal set A set of all possible outcomes of all the sets is called Universal set. It is denoted by the letter ‘U’Let A = {1,2} B= {2,3,4} C= {5,6,7,8} then Universal set can be defined as U = {1,2,3,4,5,6,7,8}Or U = {x | x ≤ 8 }

3.4 List of Symbols3.4.1 Belongs to (∈) : x ∈ A : x is an element of set A If A = set of all the capital letters of the English alphabet. X = element x has value ‘B' thenx is one of the elements of set A and can be written as x ∈ A and read as x belongs to set A.

3.4.2 Not Belongs to (∉ ) : x ∉ A : x is not an element of set AIf A= set of all capital letters of the English alphabet.x= element x is having value ‘b’then value of x is ‘b’ (i.e. small ‘b’ and not capital ‘B’) Thus, it is not an element of set A and can be written as x ∉ A and read as x does not belong to set A.

3.4.3 Equal to ( = ) : A = B : A and B are equal sets.Let A= {1,2,3} B= {1,2,3} then A = BA= {10,20,30}B= {20,30,10}even then A = B

3.4.4 Not Equal to ( ≠ ) : A≠B : A and B are unequal sets.Let A= {1,2,3} B= {10,20,30} then A ≠ BA= {10,20,30}B= {20,30,10,40} even then A ≠B

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3.4.5 Subset (⊂) : A⊂ B : Set A is a subset of set BLet A={1,2,3,4} B={1,2,3,4,5,6}Since every element of set A is also an element of set B, A is called subset of B.Symbolic representation as x ∈ B for ∀ χ ∈ A [∀: all or every]Therefore, A ⊂ B

3.4.6 Not a Subset (⊄): A ⊄ B: A is not a subset of BLet A={1,2,3,4} B={2,3,4,5,6}Since every element of set A is not an element of set B, A is not a subsetof B. Symbolic representation is as x ∉ B for x ∈ A [ : all or every ]Here, an element having value 1 is in A but it is not in set B [ 1∈A but1∉B] Thus, A ⊄ B.

3.5 Operations on Sets3.5.1 Union (∪): A ∪ B: Union set of sets A and B It contains all the elements of set A as well as set B.If A= {a, b, c, d, e}B = {d, f, g}then A∪ B = {a, b, c, d, e, f, g}

3.5.1 Intersection (∩): A ∩ B: Intersection set of sets A and BIt contains all the elements which are common to set A and set B.If A= {a, b, c, d, e} B = {d, f, g} then A ∩ B = {d }If there is no common element between the two sets A and B then the sets are called Disjoint sets. Or if A ∩ B = φ then sets A and B are called Disjoint sets.Let A= {1,2,3} B={4,5,6} then A ∩ B = φ

3.5.2 Complementary Set ( ′ or ) : A′ or A : Complementary set of A. If a set has all elements of a Universal set except the elements of set A,then that set is called the complementary set of A. It is denoted by A′ or Ac

Let U = {1,2,3,4,5} A= {1,3} then A′ or Ac = {2,4,5}

Difference ( - ) : A• −B : Different set of sets A and B. The different set of two sets A and B are the set of elements of A which are not in B and is denoted by A− B Let A = {1,2,3,4,5,6,7} B = {2,4,6} then A- B = {1,3,5,7 } Note that A − B ≠ B − A The different set of two sets B and A is the set of elements of B which are not in A and is denoted by B − A Let A = {1, 2, 3, 4, 5, 6, 7} B = {2, 4, 6} then B− A = { }= φ Example: Let A= {1, 2, 3, a} B= {1, 2, b, c, l, m} then A- B = {3, a} B- A = {b, c, l, m}


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Cartesian Product (• x): A x B: Cartesian product of sets A and B. For non-empty sets A and B, A x B is the set of all ordered pairs (a, b) of elements a∈ A , b∈ B Thus, A x B = {(a, b): a∈ A, b∈ B}Note that A x B ≠ B x AExample:Let A= {a, b, c} B = {1,2}thenA x B = {(a, 1) (a, 2) (b, 1) (b, 2) (c, 1) (c, 2) }B x B = {(1, 1) (1, 2) (2, 1) (2, 2) } Note that (1, 2) ≠ (2, 1)

3.6 De Morgan’s LawThe complement of the union of two sets is equal to the intersection of•

their complements. (A∪B)′ = A′∩B′ or (A ∪B)c = Ac ∩Bc

• The complement of the intersection of two sets is the union of their complements. (A∪B)′ = A′∩B′ or (A ∪B)c = Ac ∩Bc

Some more results:If A is a finite set, then we denote the number of elements in A by symbol n(A) or m(A)Consider A and B as finite sets then,

n(A• ∪B) = n(A) + n(B)- n(A∩B)n(A• ∪B∪C) = n(A)+n(B)+n(C)- n(A∩B)- n(B∩C)- n(A∩C) + n(A∩B∩

3.7 Application:All mathematical concepts are now defined formally in terms of sets and set theoretic concepts. For example, • mathematical structures as diverse as graphs, manifolds, rings, and vector spaces are all defined as sets having various (axiomatic) properties Equivalence and order relations are ubiquitous in mathematics, and the theory of relations is entirely grounded • in set theorySet theory is also a promising foundational system for much of mathematics, i.e. all mathematical theorems • can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic For example, properties of the natural and real numbers can be derived within set theory, as each number system • can be identified with a set of equivalence classes under a suitable relation whose field is some infinite set Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is • likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory Few full derivations of complex mathematical theorems from set theory have been formally verified, however, • because such formal derivations are often much longer than the natural language proofs mathematicians commonly present One verification project, Metamath, includes derivations of more than 10,000 theorems starting from • the ZFC axioms and using first order logic

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SummarySet theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) • of A, we write o ∈A. Since sets are objects, the membership relation can relate sets as wellA derived binary relation between two sets is the subset relation, also called set inclusion. If all the members • of set A are also members of set B, then A is a subset of B, denoted A ⊆ B There are six Concepts available in set theory. The concepts are, Union• The set A and B is symbolized by A• ∪B .That is group the values of the set A and B. IntersectionThe set A and B symbolized by A ∩B. It means we only select the common values of the set A and B. • ComplementIt is represented by A• c is the all values of U that are not components of A. DifferenceThe sets A and B are the group of all objects. In that entity that is an element of accurately one A and B. Cartesian • Product A x B is the Cartesian product of set A and B. Powers Set Whose elements are all possible subsets of A is called • the power set of A

ReferencesRajendra Akerkar, Discrete Mathematics: Set theory, Dorling Kindersley Publication India. 2008. 2• nd Impression 2009. P109-123T. Veeraranjan, Discrete Mathematics with graph theory & Combinatorics: Set theory, McGraw-Hill Publication, • 7th Edition. 2008. P51-64Seymour Lipschutz, Set theory and related topics: Set Theory, Mcgraw-Hill Publication. 1998, 2• nd Edition. P1-35

Recommended ReadingsDonald Waters(2006), Quantitative Methods for business, Set Theory, Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Set theory, Elsevier, • J. Curwin Slater(2007), Quantitaitve Methods, Set theory, Thomson Learning, •


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Self AssessmentThe Universal set U= {1,2,…..,9} an the sets:1.

A= {1,2,3,4,5}, B={4,5,6,7}, C={5,6,7,8,9}, D={1,3,5,7,9}, E={2,4,6,8} F={1,5,9}.Find out the following;

&

The Universal set U= {1,2,…..,9} an the sets:2. A= {1,2,3,4,5}, B={4,5,6,7}, C={5,6,7,8,9}, D={1,3,5,7,9}, E={2,4,6,8} F={1,5,9}.Find out the following;

&


&


&

Determine which of the following sets are finite:5. Lines parallel to the x axis,a. Letters in the English alphabetb. Months in the yearc. Animals living on the earth d.

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Find the elements of the set A=[{1,2,3}{4,5},{6,7,8}]. Determine whether each of the following is 6. True or False.

7.

8.

{{4,5}}9. ⊆ A

Consider the class A of Set = =[{1,2,3}{4,5},{6,7,8}]. Find the subclass of B of A, consists of the sets in A 10. with exactly;

Three elementsa. Four elementsb.

let A={a,b,c,d,e} B={a,b.d.f,g} C={b,c,e,g,h} D={d,e,f,g,h}. Find11. Aa. ∪B

let A={a,b,c,d,e} B={a,b.d.f,g} C={b,c,e,g,h} D={d,e,f,g,h}. Find 12. B∩C

let A={a,b,c,d,e} B={a,b.d.f,g} C={b,c,e,g,h} D={d,e,f,g,h}. Find 13. C∩D

let A={a,b,c,d,e} B={a,b.d.f,g} C={b,c,e,g,h} D={d,e,f,g,h}. Find 14. A∩D

Find A∩B when A = {11, 13, 16} and B = {7, 11, 16, 17}.15. {7, 11, 16a. }{8,11, 16}b. {11, 13, 16}c. {11, 13, 17}d.

Find 16. A∪B when A = {11, 13, 16} and B = {7, 11, 16, 17}.{7,11,13,16,17}a. {11,13,16,17}b. {7,11,16,17}c. {7,11,!7}d.


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Chapter IV

Arithmetic Progression & Geometric Progression

AimAim of this chapter is to introduce the basic concept of Arithmetic & Geometric Progression•

Objective

Objective of this chapter is to understand the following topics;

Progression/sequences•

Arithmetic & geometric progression•

Applications of A.P & G.P•

Properties of A.P & G.P•

Learning Outcome

At the end of this chapter students will be conversant with

Use of A.P. & G.P. in mathematics & its allied branches•

Able to define the arithmetic representation of series•

Able to define the geometric representation of series•

Able to apply the AP & GP in•

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4.1 Introduction:A series in which terms increase or decrease by a common difference is called Arithmetic Progression.The following series are in A.P.1+2+3+4+5+….. common difference = 13+5+7+9+11+…… common difference = 2 are series, in the first of these series, the first term is 3 and next terms are obtained by adding 2 each time the preceding term. In the second of these series each term, beginning with the second, is the double of the preceding term.Note-1:The nth term of A.P. is also called general term & denoted by, i = a+(n-1)dNote-2The rule to find the common difference (d): Subtract first term from the second term.

4.2 Arithmetic Progression:An arithmetic progression is a list of numbers where the difference between successive numbers is • constant. The terms in an arithmetic progression are usually denoted as u• 1; u2; u3 etc. where u1 is the initial term in the progression, u2 is the second term, and so on; un is the nth term.An example of an arithmetic progression is, •

2, 4, 6, 8, 10, 12, 14. Since, the difference between successive terms is constant, we have u3 - u2 = u2 - u1 and In general, un+1 – un = u2 – u1 We will denote the difference u2 - u1 as d, which is a common notation.

4.3 Geometric Progression:A geometric progression is a list of terms as in an arithmetic progression but in this case the ratio of successive terms is a constant. In other words, each term is a constant time the term that immediately precedes it. Let’s write the terms in a geometric progression as u1; u2; u3; u4 and so on.

An example of a geometric progression is10, 100, 1000, 10000.Since the ratio of successive terms is constant, we have

The ratio of successive terms is usually denoted by r and the first term again is usually written a.

4.4 Sequences4.4.1Definition

A sequence is a collection of numbers arranged in some order and obtained in succession according to some • definite ruleThe individual numbers forming a sequence are called the terms of the sequence•

ExamplesLet 2, 4, 6, 8, 10, 12, 14,……… be a sequence and 2 or 4 or 6 etc. are theterms of the sequence.


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4.4.2 nth Term of the Sequence ( t ) n

For a sequence we can find the nth term according to some definite rule used in that sequence.

Examples(1) Let 3, 6, 9, 12, 15, 18, 21……………… be a sequence. Find t . n

Here, we observe that the first term is 3 (i.e.3x1),the second term is 6 (i.e. 3x2), the third term is 9 (i.e. 3x3) etc. So, in general the nth term is 3x n that is 3n. Therefore, t = 3n n

Quantitative Methods(2) Let the sequence be 21, 22, 23, 24, 25,26,…………. Find t . n

Here, t = 2n n

If the nth term of the sequence is given, we can find the terms of the sequence in succession.

Examples(1) Let t = 2n-1, find the sequence. n

For n=1 t = 2(1)-1 = 1 1

For n=2 t = 2(2)-1 = 3 2

For n=3 t = 2(3)-1 = 5 3

For n=4 t = 2(4)-1 = 7 4

Therefore the required sequence is 1, 3, 5, 7,……………(2) Let t = 1/n, find the sequence. n

Here, the required sequence is 1/1, 1/2, 1/3, 1/4,……….

4.4.3 Sum of the First n Terms (S ) n

Consider the sequence t ,t ,t ,t ,……….t then the sum of the first n terms is 1 2 3 4 n

denoted by sn.

S = t1 + t2 + t3 +t4 + ……….t n

The sum of the first n-1 terms is denoted by Sn-1

S = t + t + t +t + ……….t n-1 1 2 3 4 n-1

The nth term can be obtained using the sum of the sequences as followst = S -S n n n-1

Example(1) If S = 3n2 – 4n, find the sequence. n

Let S = 3n2 - 4n …………………….(1) n

S = 3(n-1)2 - 4(n-1) n-1

= 3(n2 - 2n +1)- 4n+4 using (a-b)2 = a2 – 2ab+b2

S = 3n2 -10n +7 ………………….(2) n-1

Subtracting (2) from (1), we get tn

t = S -S n n n-1

Therefore, t =(3n2 - 4n)-(3n2 -10n +7) n

t = 6n- 7For n =1 t = 6(1)-7 = -1 n 1

For n =2 t = 6(2)-7 = 5 2

For n =3 t = 6(3)-7 = 11 3

For n =4 t = 6(4)-7 = 17 4

Therefore, the required sequence is -1, 5, 11,17,…………..

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4.5 Arithmetic Sequence or Arithmetic Progression4.5.1DefinitionIf for a sequence t - t is constant for all ‘n’, then it is called an arithmetic n+1 nsequence or arithmetic progression.(A.P.)The constant difference t - t is called the common difference of the A.P. and n+1 n denoted by ‘d’.In general, the terms of A.P. are a, a+d, a+2d,a+3d,………

4.5.2 nth Term of an A.P.To find the nth term of an A.P whose first term and common difference is known,we use the following formula as,t = a+(n-1)d where a= first term and d = common difference. n

Example(1) Find the n th term of -26,-23, -20 ,-17 ,………….. Since t2-t1 = -23-(-26) = 3 t3-t2 = -20-(-23) = 3 Therefore, the difference is constant and the given sequence is A.P. Here, the first term = a= -26 Common difference = d= 3 Using t = a+(n-1)d we have, n

t = -26+(n-1)(3) n

t = -26+3n-3 n

Therefore the nth term = 3n-29.

4.5.3 Sum of the First n Terms of an A.P. (S ) n

To find the sum of the first n terms of an A.P.If the first term (a) and common difference (d) is known then,S = n/2 [ 2a + (n-1)d] n

If the first term (a) and the last term of the nth term (L) is known then,S = n/2 [ 2a + L ] n

Example(1) Find the sum of n terms of A.P. -26, -23, -20, -17, ………….. Here, first term = a= -26 And common difference = d = 3 Therefore, using S = n [ 2a + (n-1)d] we get, n

2 S = n/2 [ 2(-26) +(n-1)(3)] n

S = n/2 [ n-55] n

S = n/2 (n-55) n

4.5.4 Properties of an A.P.(1) Let a, b, c, d,………be an A.P. and x a constant quantity. Then a+x, b+x, c+x,…………. a-x, b-x, c-x,…………. ax, bx, cx, …………. a/x, b/x, c/x,…………. are all in A.P.(2) When three quantities are in A.P., the middle one is called the arithmetic mean (AM) of the other two. Let a, b, c be an A.P. then the AM between them is b = a+c/2


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4.6 Geometric ProgressionIf for a sequence the ratio t /t is constant for all ‘n’, then it is called a n+1 n geometric sequence or geometric progression.(G.P.)The constant ratio t /t is called the common ratio of the G.P. and denoted n+1 n by ‘r’.In general, the terms of G.P. are a, ar, ar2, ar3,………th

4.7 nth Term of a G.P.To find the n th term of a G.P. whose first term and common ratio is known, we use the following formula ast = ar(n-1) where a = first term and r = common ratio. n

Example(1) Find the nth term of 5, 15, 45, 135, ………….. Since t2/t1 = 15/5 = 3 t3/t2 = 45/15 = 3 Therefore, the ratio is constant and the given sequence is G.P. Here, the first term = a= 5 Common ratio = r= 3 Using t = ar(n-1) we have, n

t = 5x 3(n-1) n

Therefore the n th term = t = 5x 3(n-1) n

4.8 Sum of the First n Terms of a G.P. (S ) n

To find the sum of the first n terms of a G.P.If first term(a) and common ratio(r) is known then,(i) if r¹1 and r < 1 then S = a ( 1- rn) n

1- r

(ii) if r¹1 and r > 1 then S = a ( rn - 1) n

r - 1(iii) if r =1 then The sequence a , ar, ar2, ar3, ……………… can be written as, a,a,a,a,………..n times. Therefore,S=ann

Example(1) Find the sum of n terms of G.P. 5,15,45,135,…………. Here, first term = a= 5 And common ratio= r = 3 Since r¹1 and r > 1 then Therefore, using S = a (rn - 1) we get, n

r - 1 n S = 5 [3 -1] n

3 1 S = 5 (3n-1) n

2Note(1) Sum of the series as n tends to infinity and r<1 is (S ) = a / (1-r)(2) Geometric mean of two numbers is obtained as G.M.= ab

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SummaryArithmetic progression:•

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.The following formula is used for solving the nth :tn = arn-1

where a, the first termn, number of termsr, the common ratio

Geometric progression:• A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratioFormula:tn = arn-1

where a, the first termn, number of termsr, the common ratio

Points to Remember:Arithmetic Progression:

T• n = a + (n+1)d

S• n = =

T• n = Sn - Sn-1

A.M. of a & b = •

S• n = A.M. x n

Geometric Progression:T• n= arn-1

S• n = =

Sum of the terms of an infinite G.P = •

Sum of the square of the terms which are in infinite G.P. is •

T• n = Sn – Sn-1

G.M. of a & b = •

Product of all •

n terms of a G.P. = (G.M.)• n


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ReferencesH. Dubner, Large Sophie Germain primes, Math. Comp. • 65 (1996), 393-396R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, pp. 15-18. • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, New • York, 1979 L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp. • 21 (1967), 489 H. L. Nelson, There is a better sequence, J. Recreational Math. • 8 (1) (1975), 39-43 P. A. Pritchard, A. Moran, and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. • 64 (1995), 1337-1339

Recommended ReadingsDonald Waters(2006), Quantitative Methods for business, Progression, Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Arithmetic Progression, Elsevier• J. Curwin Slater(2007), Quantitaitve Methods, Progression, Thomson Learning•

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Self AssessmentIF all the terms of an arithmetic progression are multiplied by a constant quantity te resulting terms will always 1. form;

Geometric progressiona. Arithmetic progressionb. Either arithmetic or geometric c. Both progressionsd.

Find the terms specified in the following series;2. 12a. th & 77th terms in the series 10, 12, 14, 16,……32, 162b. 172, 32c. 34, 68d. 32, 164e.

73. th & 11th terms of the series 1/3, 1/9, 1/27, 1/81, …..3a. 7 311

1/3b. 10, 1/311

1/3c. 11, 1/37

1/3d. 7, 1/311

The first term of an arithmetic progression is -15 & its 464. th term is 120, what are the values of 23rd & 6th term.54, 0a. 51, 0b. 51, 3c. 54, 3d.

If the 65. th term of an arithmetic progression is 10 & 9th term is 14.5, then find 2nd term.2.5a. 7.5b. 4c. Cannot be determinedd.

Which term of the arithmetic progression 2/3, ¾, 5/6 , …… in 23/6 ?6. 72a. 73b. 58c. 63d.

Find the sum of ten terms of an A.P. whose 57. th term is 5 & 7th term is 3.45a. 36b. 72c. 55d.


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How many integers are there between 300 & 600 that are divisible by 9?8. 1500a. 276b. 284c. 248d.

Find the first term & the common ratio of G.P. whose 69. th term & 9th are 160 & 1280 respectively?3, 5a. 10, 4b. 10, 2c. 5, 2d.

If 7/12 , -2m and 12/7 form a G.P. then what is the value of ‘m’?10.

a. 2

b. ½1/c. 1/2d.

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Chapter V

Probability

AimAim of this chapter is to introduce probability•

Objective

Objectives of this chapter are

To enlighten the students with the term of probability by providing its definitions and various •

terminologies used in probability

To explore the conditional probability and steps to solve probability•

To enable the students to solve practical problems on how to calculate probability using Addition •

theorem, multiplication theorem

Learning Outcome

At the end of this chapter students will be conversant with

Acquire basic knowledge about probability with its basic terminologies•

Enable and enrich themselves with the various approaches to probability•

Co• me out with solutions of calculating probability using Addition theorem, multiplication theorem


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5.1 Introduction:Every human activity has an element of uncertainty. Uncertainty affects the decision making process. “Probably”-this word is often used by us like probably it may snow today; probably there may be a surprise test tomorrow etc. So, there is a need to handle uncertainty systematically and scientifically. Hence, probability theory is used.

5.2Definition:“Probability is the ratio of favorable events to the total number of equally likely events.”-By Laplace• “Probability is an attitude of mind towards uncertain events.”-By Connor• Probabilities are associated with experiments where the outcome is not known in advance or cannot be • predicted

Example: If you toss a coin, will you obtain a head or tail? If you roll a die will obtain 1, 2, 3, 4, 5 or 6? • Probability measures and quantifies “how likely” an event, related to these types of experiment, will happen • The value of a probability is a number between 0 and 1 inclusive • An event that cannot occur has a probability (of happening) equal to 0 and the probability of an event that is • certain to occur has a probability equal to 1In order to quantify probabilities, we need to define the • sample space of an experiment and the events that may be associated with that experiment

5.3 Sample Space & Events:The • sample space is the set of all possible outcomes in an experimentAn • event is a subset of the sample space

Example:If a die is rolled, the sample space S is given by,S = {1, 2, 3, 4, 5, 6}

If two coins are tossed, the sample space S is given byS = {HH,HT,TH,TT} , where H = head and T = tail.

5.4 The Venn Diagram:A venn diagram is a pictorial presentation of the sample space of an experimentIt is usually drawn as a rectangular figure representing the sample space & it contains circles or other shapes representing events in the sample space Heads Tails Venn diagram representing outcomes of tossing a coin.

A B Venn diagram representing outcomes of selecting a manager. A= candidate has over 3yrs experience. B= candidate has post graduate qualification.

5.5 Rules of Probability:A probability is number assigned to the occurrence of an event in sample space.• A probability of an event must be greater than or equal to 0 & less than or equal to 1 or 100% i.e. • 0• means probability cannot be negative.If A & B are mutually exclusive events, then the probability of (A or B) is equal to the sum of the probabilities • of A & B.

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5.6 Applications:Probability used in risk assessment at any organizational level, especially at top management where the crucial • decisions are takenIt is also applied to the commodity markets in trading• Governments typically apply probabilistic method in Environmental regulation where it is called Pathways • analysisIt is also applicable where consumer products, such as automobiles & consumer electronics, utilize reliability • theory in the design of the product in order to reduce the probability of failure, the probability of failure may be closely associated with the product’s warranty

5.7 Conditional Probability:5.7.1 Independent & Dependent Events:Two events A & B are independent events if the occurrence of event A is in no way related to the occurrence or non-occurrence of event B.Likewise for independent events the occurrence of event B is in no way related to the occurrence of event A.

5.7.2 Multiplication Rule:The joint probability of two independent events is equal to the product of their marginal probabilities.P(A and B) = P(A).P(B)

5.7.3 Conditional Probability: Independent eventsIf the probability of an event is subject to a restriction on the sample space, the probability is said to be conditional probability.

We define the conditional probability of event A, given that B has occurred, in case of A & B being independent events, as the probability of event A.P(A|B) = P(A).

5.7.4 Conditional Probability: Dependent EventsWe define the conditional probability of event A; given that event B occurred when both A & B are dependent events, as the ratio of number of elements in B.P (A|B) = .

5.7.5 Multiplication Rule: Dependent EventsA joint probability of two dependent events A & B is equal to probability of A multiplied by probability of B, given that A has occurred.

P(A and B) = P(A).P(A|B)

This formula is derived from the formula of conditional probability of dependent events.

P(B|A) =

P(A and B) = P(B|A).P(A)


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5.8 How to calculate the Probabilities?Classical Probability Formula: • It is based on the fact that all outcomes are equally likely •

Total number of outcomes in EP(E) = ________________________________________________ Total number of outcomes in the sample space

Empirical Probability Formula: • It uses real data on present situations to determine how likely outcomes will occur in the future. Let us clarify this using an example: 30 people were asked about the colors they like and here are the results:

Color frequencyred 10blue 15green 5

If a person is selected at random from the above group of 30, what is the probability that this person likes the red color? Let event E be “likes the red color”. Hence,

Frequency for red colorP(E) = _______________________________________________ Total frequencies in the above table

P (E) = 10 / 30 = 1 / 3

5.9 Steps to solve probabilityTo solve any problem on probability the steps involved are

Define the events• Find the total outcome of the experiment• Find the probability of each event• If the words “either, or” are used check whether the events are mutually exclusive or not to apply addition • ruleIf the words “both, and” are used check whether the events are independent or dependent to apply proper • multiplication ruleTo find the total outcome of the experiment use 2• n or 6n in the case of coin or dice respectively, where n is the number of coins or dice thrown at a time or a coin or dice thrown n times. In all other cases, use nCr =

For example,

6C3 =

Example 1: What is the chance of getting a King in a draw from a pack of 52 cards?Answer:The total no. of cards = 52The total no. of Kings = 4

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Bayes Theorem:Statement:Thomas Bayes addressed both the case of discrete probability distributions of data and the more complicated case of continuous probability distributions. In the discrete case, Bayes theorem relates the conditional and marginal probabilities of events A and B, provided that the probability of B does not equal to zero.Each term in Bayes’ theorem has a conventional name:

P(• A) is the prior probability or marginal probability of A. It is “prior” in the sense that it does not take into account any information about B.P(• A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from or depends upon the specified value of B.P(• B|A) is the conditional probability of B given A. It is also called the likelihood.P(• B) is the prior or marginal probability of B, and acts as a normalizing constant.

Bayes’ theorem in this form gives a mathematical representation of how the conditional probability of event A given B is related to the converse conditional probability of B given A.

P (A|B) = P (B|A).P(A) P (B)


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SummaryProbability theory is a branch of mathematics concerned with the analysis of random phenomena. The • outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chanceThe probability of an event is to find out how many times event will happen because of a research. They will • give a clearly results. The probability is the study of chance or possibility of an event to happening like straight or not directly, probability plays a role in the all activitiesAs a mathematical foundation for statistics, probability theory is essential to many human activities • that involve quantitative analysis of large sets of dataThe probability of event A is the number of ways event A can occur divided by the total number of • possible outcomesDiscrete probability theory deals with events that occur in countable sample spaces whereas continuous • probability theory deals with events that occur in a continuous sample space

ReferencesCharles Miller Grinstead, James Laurie Snell, Introduction of Probability: Probability, AMS Bookstore, 1997, • p133-137Frederick Mosteller, Probability: • Probability, Dover Publications, 1987, First Edition, p1-100Dartmouth, Probability http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/• book.pdf, last accessed: 18 oct 2010Math Goodies, Probability, http://www.mathgoodies.com/lessons/vol6/intro_probability.html, last accessed: • 18 oct 2010Britannica Encyclopedia, Probability, http://www.britannica.com/EBchecked/topic/477530/probability-theory, • last accessed: 18 oct 2010

Recommended ReadingsDonald Waters(2006), Quantitative Methods for business, Probability , Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Probability, Elsevier • J. Curwin Slater(2007), Quantitaitve Methods, Probability , Thomson Learning,•

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Self AssessmentIf one event is unaffected by the outcome of another event, the two events are said to be;1.

Dependenta. Independentb. Mutually exclusivec. All of the above.d.

The probability that the sum 8 appears in asingle toss a pair of fair dice is,2. 4/36a. 5/36b. 6/36c. 7/36d.

If P(A or B) = P(B)3. A & B must be mutually exclusivea. A & B must be independentb. P(A)=P(B)c. Occurrence of A implies occurrence of B.d.

For any two events A & B, which of the following is TRUE?4. P(A or B) = P(A) + P(B) – P(A and B)a. P(A or B) = P(A) + P(B)b. P(A and B) = P(A) x P(B)c. P(A or B) = P(A) - P(B)d.

A fair coin is tossed twice. The probability that two heads will occur is;5. 0.15a. 0.25b. 0.375c. 0.50d.

If P(A or B) = P(A) + P(B) the A & B are;6. Independenta. Dependentb. Conditionalc. Mutually exclusived.

Which of the following is not correct?7. P(A/B) = P(A and B)/P(B)a. P(A and B) = P(B/A).P(A)b. P(B or A) = P(B). P(A/B)c. P(B and A) = P(A/B).P(B)d.


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Bayes theorem is formula for 8. Conditional probability under statistical dependencea. Conditional probability under statistical independenceb. Unconditional probability under statistical dependencec. Unconditional probability under statistical independenced.

If P(A), P(B) & P(B/A) are 0.4, 0.5, 0.7 respectively, then P(A/B) is;9. 0.140a. 0.560b. 0.320c. 0.286d.

If P(AB) = 0.8 and P(A) = 0.40 then the probability of occurrence of event B given that event A has occurred 10. is;

0.21a. 0.20b. 0.30c. 0.31d.

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Chapter VI

Permutation & Combination

AimAim of this chapter is to introduce the concept of permutation & combination•

Objective


To explain the fundamental principle of counting•

To enlighten the students with permutation concept & its uses•

To explicate the complementary combinations & its use in probability•

To elucidate the applications of permutation & combination•

Learning Outcome

At the end of this chapter students will be acquainted with the following

Understand the concept of permutations and combinations•

Distinguish between permutation and combination•

Understand factorial notation•

Understand the meanings of • nPr and nCr

Develop the skill in solving the problems•


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6.1 Introduction to principles of counting:

If one operation can be performed in m ways and And a second operation can be performed in n ways then both • the operations could be performed simultaneously or sequentially in mxn ways

Addition Principle: If One Operation can be performed in m ways and another operation can be performed in •

n ways then one of the two operations could be performed in m+n ways

For Example, a person wants to borrow a book from the library. There are Five novels and Six story books •

which he wants to read. He could select any one of the min m+n=5+6=11 ways

6.2DefinitionofPermutation:Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.Note: The sign ‘!’ indicates the factorial notation. It can be calculated as

n! = n(n-1)! Or n! = n (n-1) (n-2)! Or n! = n(n-1)(n-2)…………1Where 0! = 1 and 1! = 1For example it can be calculated as,3! = 3 x 2 x 1 = 64! = 4 x 3 x 2 x 1 = 245! = 5 x 4 x 3 x 2 x 1 = 120

6.3 Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.The total number of these combinations is denoted by nCr and calculated as ,

Example:Suppose we have to form a number of consisting of three digits using the digits • 1,2,3,4, To form this number the digits have to be • arranged. Different numbers will get formed depending upon the order in which we arrange the digits.This is an example • of Permutation.Now suppose that we have to make a team of 11 players out of 20 players, This is an example of • combination, because the order of players in the team will not result in a change in the team. No matter in which order we list out the players the team will remain the same! For a different team to be formed • at least one player will have to be changed.

6.4 Fundamental principles of counting:6.4.1 Addition Rule :If an experiment can be performed in ‘n’ ways, & another experiment can be performed in ‘m’ ways then either of the two experiments can be performed in (m+n) ways. This rule can be extended to any finite number of experiments.

6.4.2 Multiplication Rule: If a work can be done in m ways, another work can be done in ‘n’ ways, then both of the operations can be performed in m x n ways. It can be extended to any finite number of operations.

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Factorial n: The product of first ‘n’ natural numbers is denoted by n! n! = n(n-1) (n-2) ………………..3.2.1.Example: 5! = 5 x 4 x 3 x 2 x 1 =120

Note: 0! = 1Proof n! =n, (n-1)! Or (n-1)! = [n x (n-1)!]/n = n! /n Putting n = 1, We have, O! = 1! /1 or 0 = 1.

Note that:

An important result: The fundamental principle of counting (F.P.C) states that if an operation can be performed in m different ways • and if for each such choice, another operation can be performed in n different ways, then both operations, in succession can be performed in exactly mn different ways. The principle can also be generalized, for even more than two operations.For n Î N, the factorial of n is defined as n! = 1 ´ 2 ´ 3 ´ ..... ´ n. 0! is defined as 1.• The arrangements of a number of things taking some or all of them at a time are called permutations. The total • number of permutations of n distinct things taking r(1 £ r £ n) at a time is denoted by nPr or by P(n, r).For 1 £ 4 £ n, • nPr = n(n - 1)(n - 2)...... r factors.In particular, • nPn = n(n - 1)(n - 2).....n factors.= n(n - 1)(n - 2)...... 3.2.1. = n!• If p• 1 objects are of first kind and p2 objects are of the second kind, then the total number of permutations of all the p1+p2 objects is given by

If p• 1 objects are of the ith kind and i = 1,2,3,….r, then the total number of permutations of all the p1+p2+p3+.......+pr objects is given by,

The number of permutations of n different things taking r at a time when each thing is allowed to repeat any • number of times in any arrangement is given by nr.The number of circular permutations of n different things is given by (n - 1)!.• If the number of circular permutations of n different things when an anticlockwise circular permutation and its • corresponding clockwise circular permutation are considered as same circular permutation, then the number of circular permutations is

The selections (groups) of a number of things taking some or all of them at a time are called combinations. The • total number of combinations of n distinct things taking r(1£ r £ n) at a time is denoted by nCr or by C(n, r).


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SummaryPermutaions:The different arrangements of a given number of things by taking some or all at a time, are called permutation.Number of permutation- number of all permutations of n things, taken r at a time is given by;

Combiantions:Each of the different groups or selections which can be formed by taking some or all of a number of objects, is called a combination.Number of combinations: the number of all combinations of n things, taken r at a time is;

ReferencesKrishna R. Kumar, Discrete Mathematics, Permutation & Combination, Firewall Media, 2005, p23-47.• G. Shankar Rao, Discrete mathematical structure, Permutation, New Age International, 2002, p14-54.• R.C. Pennar, Discrete Mathematics, Combination, World Scientific, 1999, p54-67.•

Recommended ReadingsDonald Waters(2006), Quantitative Methods for business, Permutation & Combination, Prentice Hall • Publication,4th EditionDiana Bedward (1999), Quantitative methods, Permutation, Elsevier,.• J. Curwin Slater(2007), Quantitative Methods, Combination ,Thomson Learning.•

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Self AssessmentEvaluate the following:

1.

860a. 870b. 880c. 885d.

602. 204020a. 205320b. 215320c. 225320d.

3. 24a. 53b. 23c. 22d.

How many words can be formed by using all the letters of the word ‘BIHAR’?4. 121a. 120b. 123c. 130d.

In how many ways can a cricket eleven be chosen out of batch of 15 players?5. 1360a. 1365b. 1370c. 1375d.

How many words can formed from the letters of the word ‘DIRECTOR’ so that the vowels are always 6. together?

2100a. 2121b. 2160c. 2170d.

In how many different ways can the letters of the word ‘RUMOUR’ be arranged ?7. -180a. 90b. 30c. 720d.


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From a group of 7 men & 6 women, five persons are to be selected to form a committee so that at least 3 men 8. are there on the committee. In how many ways can it be done?

564a. 645b. 735c. 756d.

In how many different ways can the letters of the word ‘MACHINE’ be arranged so that the vowels may occupy 9. only the odd positions?

210a. 576b. 144c. 1728d.

In how many ways can a group of 5 men & 2 women be made out of total of 7 men & 3 women?10. 266a. 5040b. 11760c. 86400d.

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Chapter VII

Interpolation

AimAim of this chapter is to introduce the concept of Interpolation•

Objective

Objective of this chapter are;

To enlighten the different methods of interpolation•

To describe interpolation techniques to calculate the desired figures•

To entails the degree of dependent & independent variables which helps in calculation & methods of •

interpolation

Learning Outcome

At the end of this chapter students are able to understand the following;

Meaning of Interpolation•

Methods of Interpolation•

Application of interpolation in management & other sectors•


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7.1 Introduction:Interpolation is the method of statistical estimation & the word literally means making insertions.Simply interpolation is understood by following example;If we need to know the population of our country, for any intermediary year, say 1985, one logical approach would be to work forward from the population of 1981, by adding births & inflow of the people into the country & deducting deaths & outflow of people from the country during 1981-1985.Thus the data on population of the year 1985 is required, 100% accurate figures are really not required.

7.2DefinitionofInterpolation:“Interpolation” may be defined as the technique of obtaining the most likely estimates of certain quantity under • certain assumptions. - D.N. ElhanceInterpolation is a statistical device used to estimate the most likely figure under certain assumptions within the • given limitsInterpolation provides us the missing quantity of a series so that we can establish the while extrapolation are • the techniques of obtaining the most likely estimates of certain quantity under certain assumptionsInterpolated figures are not perfect substitutes of the original figures. They are only best possible substitutes • on certain hypothesis

7.3 Application:Interpolation is widely used by businessmen, administrators, sociologists, economist & financial analysis.• It helps in completing the incomplete, lost or destroyed records.• Eg. In financial analysis the interpolation used to find out the IRR(internal rate of return) of a project, all • investment decisions which require to use of the Present value & future value interest factor tables.

7.4 Need & Importance of InterpolationInadequacy of data:•

Sometimes it is not possible to collect the whole data about the problem under study. Even if it were �possible to collect the whole data it may not be worthwhile to do so due to a large amount of expenditure involved or due to organizational difficultiesTechnique of interpolation can be used for making best estimates, at the least cost �These estimates will be a more useful figure than rough estimates �

To estimate intermediate value:• In certain cases data is collected after long intervals �For example; in India the census of population is collected after every ten yrs �The technique of interpolation will be needed to estimate the figure of population for intermediate �years

Lost Data:• Sometimes the data is lost due to fire, earthquake etc. The interpolation technique is help to fill the gaps �in statistical information due to lost data

Uniformity of Data:• Statistics concerning a particular phenomenon are collected by different agencies , it destroys its �uniformity.In such cases comparison of data becomes difficult. So to establish uniformity of data, the techniques �of interpolation is used.

Forecasting:• The forecasting activity regarding data is practical utility for economic planning, policy formulation, �production decisions etc.

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7.5 Methods of Interpolation:7.5.1 Graphical Method:

It is simplest method of Interpolation.• In this method the data is represented in graph, i.e. on X-axis all independent variables & on Y-axis all dependent • variables are taken.The curve is formed after joining the points, this curve give interrelation between two variables.• From the point of X-axis, for which the value of y is to be interpolated, a line parallel to Y-axis will be drawn.• From the point where this line will cut the curve, a line parallel to X-axis will be drawn, here the value of y will • be found from the point where the line cuts Y-axis, this is called InterpolatedfigureorValue

7.5.2 Newton’s method of advancing differences:

This method is applicable for the following cases• The independent variable advances by equal intervals• The value to be interpolated is different from the equidistant value• The value to be interpolated lies in the beginning of the data• This method is known as finite or advancing differences method because after finding out differences in the • values of y, the process is extended further till only one difference remainsWe also taken into consideration that the +ve & -ve sign while calculating the differences•

7.5.3 Lagrange’s Method:

Lagrange’s interpolating polynomial is another very good formula for interpolation• This method has no restriction on the x-variable whether it should be equally spaced or not• This method can be used for any value of x either for interpolation• It is also to estimate the argument of x for given value of y, it means the Lagrange;s formula can be used for inverse • interpolation also.The only demerit of Lagrange’s formula is that, it required heavy computational work

7.5.4 Newton-Gauss Foreword Method:It is method which is used in particular situation. It is used when the independent variable (a) advance by equal intervals (b) the value to be interpolated falls in the middle of the series. The formula is as under;

+ + + +

Where X =

7.5.5 Newton-Guass Backward Method:This is also known as Newton Gregory backward formula• Its applicability is described below• When the argument x advances with the equal jumps• When the x-value to be interpolated lies near the end of the series• Diagonal difference table is used in Newton’s backward formula, but the differences are used in reverse • order.


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The formula is as under;If there are n arguments & n corresponding entries, Newton’s backward formula for the entry to be interpolated for the argument x is,

=

Where,

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SummaryAlgebraic expressions in which the variables concerned have only non-negative integral exponents are called • polynomialsThe standard form of a polynomial in one variable is that in which the terms of the polynomial are written in • the decreasing order of the exponents of the variableInterpolation provides us the missing quantity of a series so that we can establish the while extrapolation are • the techniques of obtaining the most likely estimates of certain quantity under certain assumptionsMethods of Interpolation are• Graphical method, Lagrange’s Method, Newton-Gauss Forward Method, Newtons-Gauss Backward method. • etc.

ReferencesTR Jain, AS Sandhu, Quantitative Methods: Interpolation, VK Publication, 2006-07, p7.1-7.38• B.R.Agarwal, Programmed Statistics, Interpolation, New Age International, 2• nd Edition, 2007, p405-425.N.P Bali, P.N.Gupta, A textbook of Quantitative Techniques, Interpolation, Laxmi Publications, 1• st edition, 2008, p134-145.

Recommended ReadingsWaters Donald(2006), Quantitative Methods for business, Interpolation, Prentice Hall Publication,4th • EditionBedward Diana(1999), Quantitative methods, Interpolation, Elsevier, • J. Curwin & Slater(2007), Quantitaitve Methods, Interpolation of Polynomials, Thomson Learning,•


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Self AssessmentGraphical Interpolation method is,1.

Simplea. Algebricalb. Fully reliablec. Reliabled.

Interpolation is helpful in estimating:2. A seriesa. An intermediary value of given argumentb. Entry of alternative valuesc. A series of valued.

Lagrange Formula is useful for ;3. Interpolationa. Arithmetic functionsb. Inverse rangec. Inverse extrapolationd.

Lagrange’s polynomials interpolation can be used even if;4. The given arguments are not equally spaceda. Extrapolation is to be doneb. Inverse interpolation is to be donec. Relation to be mappedd.

Interpolation formulae are based on the fundamental assumptions that the data can be expressed as;5.

A linear functiona. A quadratic functionb. A polynomial functionc. A binomial function d.

If 6. is constant then may be;Constanta. At equal intervalsb. Both a & bc. At un equal interivalsd.

The problems of interpolation are simpler than prediction because;7. Interpolation has fewer restriction than predictiona. Interpolation is based on more stringent restriction than predictionb. There are no restriction than interpolationc. It is easier to find outd.

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A finance company offers to give Rs. 100000after 10 yrs, if Rs.20000 is deposited today. Calculate the implicit 8. rate of interest using the tables.

5%a. 10%b. 12.45%c. 17.45%d.

A finance company offers to give Rs.18000 annually for 15 yrs. If a deposit of Rs.90000 is made now. Calculate 9. the implicit rate of interest using the tables.

5.092%a. 32.33%b. 9.08%c. 18.42%d.

A project requires an initial outlay of Rs.35 lakh & has the following cost flow projections:10.

Year Cash Flows (in Lakh Rs)0 351 152 103 104 20

Using interest tables & interpolation techniques find out the IRR of the project.12.11%a. 19.81%b. 13.01%c. 35.05%d.


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Chapter-VIII

Commercial Arithmetic

AimAim of this chapter is to introduce & represent the elementary principles of commercial arithmetic related to • commercial world.

Objective


To understand the terms such as Commission & Brokerage•

To understand the use of Present Worth, Sum Due & True Discount•

To describe the concept of Insurance•

To infuse a confidence about tracking the various mercantile & business problems that encounter in examination •

& real life as well

Learning Outcome

At the end of this topic students will able to use the following;

Use of present Worth, Sum Due & True Discount in the Commercial World•

Application of the commission & brokerage in different sectors of economy•

Application of arithmetic terms in insurance•

Able to apply the arithmetic topics in real life & business•

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8.1 Introduction:A good deal of business is done by means of agents. An agent can be defined as a person appointed to transact business in the name & for the account of another. His remuneration is usually in the form of an allowance on the amount of the business transacted, which is fixed by specific agreement, or in accordance with the usage of the trade. This remuneration is generally called as the agents commission & in most cases is a percentage of the amount of the transaction.

8.2 Commission & Brokerage:Commission/brokerage means any payment received by a person acting on behalf of another person or for any • services in the course of buying/selling of goods or in relation to any transaction relating to any asset.For earning commission or brokerage, three parties should be present in a transaction, i.e. the Buyer, the seller • and the broker/commission agent. The person receiving income as commission/brokerage should not own the subject matter dealt with by him • and the ownership should vest with his principal only.

8.2.1 Application of Commission:

Commission is applied to following agents;• Real Estate Agents: The Agents dealing in selling real estate properties are real estate agents.• Canvassing Agent: These are the persons who sells articles as books, aluminum ware from house to house.• Producer• Travelling Salesmen• Retail Clerks• Employment Agencies•

8.3Profit&Loss:Sales Price (Sp): The price at which goods/services are sold.• Cost Price (SP): The expenses occurred in making a product (or providing a service) and it includes the price • of the raw material.Revenue: is compensation received for your product or services. • Cost of Goods Sold: are the variable expenses related to the sale of your product. • Operating Expenses: are fixed expenses, such as rent, and utilities. • Operating Income: is profit after operating income. • Earnings before Taxes: is income including other income and expenses, but before taxes. • Income Taxes: are federal, state, and local taxes. • Net Earnings: is the profit earned by the business, and it includes all expenses, including taxes.•


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8.4 Formulae:If the shop sells a thing for more than they paid for it, then the difference is the ‘profit’: • Profit = Selling price - Cost priceIf the shop sells a thing for less than they paid for it, then the difference is the ‘loss’: • Loss = Cost price - Selling priceBoth profit and loss can be expressed either in dollars, OR as a percentage of the cost price: • Profit Percentage = (Profit / Cost Price) x 100Occasionally profit is also mentioned in terms of sales price i.e.: • Profit on selling price= (Profit / Sales Price) x 100

8.5 Interest:8.5.1 Interest:

It is the price paid for the use of borrowed money, or, money earned by deposited funds. • Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, • major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money.•

8.5.2 Simple Interest:It is the interest calculated on the original principal alone for the time during which money lent is being used. If ‘P’ is the principal, ‘R’ is the rate of interest & ‘T’ is the period, then simple interest (SI) is given by the formula, SI = PRT/100.

8.5.3 Compound Interest:When the interest produced after each prefixed period is added to the principal & whole amount is considered as principal for the calculation of interest for the next period, then the sum by which the original principal is increased at the end of all prefixed period is called compound interest.

8.6 Present WorthDefinition:Present Worth is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk.

Present Worth can be calculated by following formula;

Where, PV = present worthC= cash flowi= interest rate

8.7 True Discount (T.D.):The true discount is the difference between the sum due at the end of the given time and its present worth.Thus, True discount (T.D.) = interest on present worth.

8.8 Sum Due (S.D.):Sum due is calculated by adding Present Worth with True Discount.Sum Due (SD) = Present Worth (PV) + True Discount (TD).

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8.9 Insurance:Insurance• is defined as the equitable transfer of the risk of a loss, from one entity to another, in exchange for payment.An • insurer is a company selling the insurance; An • insured or policyholder is the person or entity buying the insurance policy.The • insurance rate is a factor used to determine the amount to be charged for a certain amount of insurance coverage, called the premium.


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SummaryCommission & brokerage-

Commission/brokerage means any payment received by a person acting on behalf of another person or for any • services in the course of buying/selling of goods or in relation to any transaction relating to any asset.

Profit&Loss-If selling price is greater than cost price, the seller is said to have a profit.• If selling price is less than cost price, the seller is said to have incurred a loss.• Cost price= it is the price at which an article is purchased.• Selling Price= it is price at which an article is sold.•

Formulae:Gain= SP –CPLoss = CP – SP

Gain % =

Loss % =

Present Worth =

An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. Present Worth is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk.The true discount is the difference between the sum due at the end of the given time and its present worth.Sum due is calculated by adding Present Worth with True Discount.

ReferencesG.R. Veena, Business Mathematics, Commercial Arithmetic, New Age International Publishers, 2006, p215-• 255, 268-270R.S Aggarwal, Quantitative Methods, S.Chand Publications, 2008, p251-293 & 632-636• Oscar Williams, Commercial Arithmetic, Profit & Loss Bibliobazar, 2008, p3-85• S. Jackson, Commercial Arithmetic, Special Application, 2009, p23, 45-67.• Geo H. Douglas, Modern Commercial Arithmetic, Business Mathematics, Commision agent, 2007, p34-90•

Recommended ReadingsJohn Henry Moore, New Commercial Arithmetic, Bibliobazaar LLC, 2008, p50-145.• Augustus D morgan, Elkements of Arithmetic, Taylor & Walton, 1900, p125-150.• Frederick calder, Elementary rules of Arithmetic, revised edition, 1852, p106, 171.•

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Self AssessmentA man purchased a cow for Rs. 3000 and sold it the same day for Rs. 3600, allowing the buyer a credit of 2 1. years. If the rate of interest be 10% per annum, then the man has a gain of:

0%a. 5%b. 7.5% c. 10%d.

The true discount on Rs. 2562 due 4 months hence is Rs. 122. The rate percent is:2.

a. 12% b. 33 % c. 15% d. 14%

A trader owes a merchant Rs. 10,028 due 1 year hence. The trader wants to settle the account after 3 months. 3. If the rate of interest 12% per annum, how much cash should he pay?

Rs. 9025.20a. Rs. 9200b. Rs. 9600.c. Rs. 9560d.

If Rs. 10 be allowed as true discount on a bill of Rs. 110 due at the end of a certain time, then the discount 4. allowed on the same sum due at the end of double the time is:

Rs. 20.a. Rs. 21.81b. Rs. 22c. Rs. 18.33d.

A man wants to sell his scooter. There are two offers, one at Rs. 12,000 cash and the other a credit of Rs. 12,880 5. to be paid after 8 months, money being at 18% per annum. Which is the better offer?

Rs. 12,000 in casha. Rs. 12,880 at creditb. Both are equally good c. 1800 at creditd.

Find the present worth of Rs. 930 due 3 yeears hence at 8% per annum. Also find the discount.6. 700a. 750b. 800c. 540d.

The true discount on a bill due 9 months hence at 12% per annum is Rs.540. Find the amount of the bill and 7. its present worth.

6250a. 6000b. 6010c. 6015d.


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A man buys an article for Rs. 27.50 and sells it for Rs. 28.60. find his gain percent.8. 5%a. 4%b. 6%c. 5.2%d.

A dishonest dealer professes to sell his goods at cost price but uses a weight of 960gms for a kg. weight. Find 9. his gain percent.

4 a. 3 b. 5 c. 6 d.

An article is sold at a certain price. By selling it at 2/3 of that price one losses 10%. Find the gain percent at 10. original price.

30 %a. 35 %b. 42 %c. 37 %d.

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Chapter-IX

Relations & Functions

AimAim of this chapter is to introduce the mathematical concept of relation & function•

Objective


To understand relation & functions•

To identify domain & range in ordered pairs•

To evaluate functions•

To create or find a real world example of domain & range•

Learning Outcome

Attheendofthischapter,studentswillbeconversantwiththefollowing

Defining the relation & its domain and range•

Defining the function & its domain & range•

Difference between relation & function•


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9.1 Relation:A relation is just a set of ordered pairs. There is absolutely nothing special at all about the numbers that are in a relation.In other words, any bunch of numbers is a relation so long as these numbers come in pairs.In maths Relation is just a set of ordered pairs.Note: {} is the symbol for ‘SET’.Example: {(0, 1), (55, 22), (3,-50)}

9.2 Domain & Range of a Relation:The Domain is the set of all the first numbers of the ordered pairs.In other words, the domain is all of the x-values.The Range is the set of the second numbers in each pair, or the y-values.

Example:ifRelationis{(0,1),(55,22),(3,-50)},thenDomain is {0 55 3}Range is {1 22 -50}

NOTE:whenwritingthedomain&range,donotrepeatthevalues

Relation can be written in several ways; Ordered Pairs• Table• Graph/mapping.•

Examples:What is the domain and range of the following relation? {(-1, 2), (2, 51), (1, 3), (8, 22), (9, 51)} Ans: Domain: -1, 2, 1, 8, 9 Range: 2, 51, 3, 22, 51

What is the domain and range of the following relation?{(-5,6), (21, -51), (11, 93), (81, 202), (19, 51)} Ans: Domain: -5, 21, 11, 81, 19 Range: 6, -51, 93, 202, 51

9.3 Functions:A function is a relationship between two sets of numbers. We may think of this as a mapping; a function maps a number in one set to a number in another set. Notice that a function maps values to one and only one value. Two values in one set could map to one value, but one value must never map to two values: that would be a relation, not a function.Example: If we write (define) a function as:f(x) = x2 then we say: ‘f of x equals x squared’ and we have,f( - 1) = 1f(1) = 1f(7) = 49f(1 / 2) = 1 / 4f(4) = 16 and so on.

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9.3.1Range,image,co-domainIf D is a set, we can say,

, which forms a new set, called the range of f.

D is called the domain of f, and represents all values that f takes.In general, the range of f is usually a subset of a larger set. This set is known as the co-domain of a function. Example: With the function f(x) = cos x, the range of f is [-1, 1], but the co-domain is the set of real numbers.

NotationsWhen we have a function f, with domain D and range R, we write:

If we say that, for instance, x is mapped to x2, we also can add

Notice that we can have a function that maps a point (x, y) to a real number, or some other function of two variables.We have a set of ordered pairs as the domain. Recall from set theory that this is defined by the Cartesian product. If we wish to represent a set of all real-valued ordered pairs we can take the Cartesian product of the real numbers with itself to obtain

.

When we have a set of n-tuples as part of the domain, we say that the function is n-ary (for numbers n=1,2 we say unary, and binary respectively).

9.4 Break Even Analysis:Break-even analysis is a technique widely used by production management and management accountants. • It is based on categorizing production costs between those which are “variable” (costs that change when the • production output changes) and those that are “fixed” (costs not directly related to the volume of production).Total variable and fixed costs are compared with sales revenue in order to determine the• levelofsalesvolume,sales value orproduction atwhich thebusinessmakesneither aprofitnor a loss (the “break-evenpoint”).In its simplest form, the break-even chart is a graphical representation of costs at various levels of activity shown • on the same chart as the variation of income (or sales, revenue) with the same variation in activity.The point at which neither profit nor loss is made is known as the “break-even point” and is represented on the • chart below by the intersection of the two lines:

NOTE: The break Even point is the point where the revenue from sales is equal to the cost of production.

For Calculating Total cost, we should know that,Profit(P)=Revenue(R)–Cost(C).Where,Total Cost = Fixed cost + Variable Cost.


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SummaryA relation is just a set of ordered pairs. There is absolutely nothing special at all about the numbers that are in • a relation.In other words, any bunch of numbers is a relation so long as these numbers come in pairs.• The • Domain is the set of all the first numbers of the ordered pairs, & The Range is the set of the second numbers in each pair, or the y-values.A • function is a relationship between two sets of numbers. Two values in one set could map to one value, but one value • must never map to two values: that would be a relation, not a function.Break-even analysis is a technique widely used by production management and management accountants. •

ReferencesTR. Jain, Quantitative Methods, Relation & Functions, FK Publication, 2• nd Edition, p54Author Stream, Relation & Functions, www.authorstream.com/.../sadamava-373982-2-1-relations-functions-• ppt-relationsfunctions-powerpoint-education/, Last accessed-10th October 2010.Tutor Vista.Com, Relation & Functions,• www.tutorvista.com › Math › Algebra, Last accessed-10th October 2010.Purple Math, Relation & Fuinctions, www.purplemath.com/modules/fcns.htm• , last accesed-10th Ocober 2010.

Recommended ReadingsDonald Waters(2006), Quantitative Methods for business, Relation & fuinction, Prentice Hall Publication,4th • EditionDiana Bedward (1999), Quantitative methods, Relation & Functions, Elsevier.• J. Curwin Slater(2007), Quantitaitve Methods, Relation & Functions, Thomson Learning.•

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Self AssessmentWhat is the domain and range of the following relation? 1.

{(-1, 2), (2, 51), (1, 3), (8, 22), (9, 51)} domain:______ range:______

What is the domain and range of the following relation? 2. {(-5, 6), (21, -51), (11, 93), (81, 202), (19, 51)} domain:______ range:______

Which relations below are functions? _________ 3. Relation #1 {(-1, 2), (-4, 51), (1, 2), (8,-51)} a. Relation #2 {(13, 14), (13, 5) , (16,7), (18,13)} b. Relation #3 {(3, 90), (4, 54), (6, 71), (8, 90)}c. Relaton #4 {(1,3)(2,4)(2,5)(3,4)}d.

Which relations below are functions? _______________ 4. Relation #1 {(3,4), (4,5), (6,7), (8,9)} a. Relation #2 {(3,4), (4,5), (6,7), (3,9)} b. Relation #3 {(-3,4), (4,-5), (0,0), (8,9)} c. Relation #4 {(8, 11), (34,5), (6,17), (8,19)}d.

Which relations below are functions? _______________ 5. Relation #1 {(3,4), (4,5), (6,7), (3,-9)} a. Relation #2 {(3,4), (4,5), (6,7), (5,4)} b. Relation #3 {(0,4), (4,-5), (0,0), (8,9)} c. Relation #4 {(8, 11), (34,5), (6,17), (6,19)}d.

For the following relation to be a function, 6. X cannot be what values? {(8, 11), (34,5), (6,17), (X ,22) }

8, 34, 6a. 11, 5, 17b. 8, 34, 6, 11, 5, 17, 22c. 8, 34, 6, 11, 5, 17d.

For the following relation to be a function, 7. X can not be what values? {(12,14), (13,5) , (-2,7), (X,13)}

12, 13, -2a. 14, 5,7,13b. 12,13,-2,14,5,7,13c. 12,13,-2,14,5,7d.


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For the following relation to be a function, 8. X can not be what values? {(12, 13), (-11, 22), (33, 101), (X ,22)}

12, -11, 33a. 13, 22, 101b. 12, -11, 33, 13, 22, 101, 22c. 12, 33d.

Suppose the weights of four students are shown in the following table.9.

Student 1 2 3 4

Weight 120 100 150 130

Find domain {1, 2, 3, 4}a. {120,100,150,130}b. {2,3,4}c. {1,3,5}d.

Suppose the weights of four students are shown in the following table.10. Student 1 2 3 4Weight 200 190 100 100

Find Range.{200,190,100,100}a. {200,100}b. {200.190}c. {200.190.100}d.

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Chapter X

Statistics

AimAim of this chapter is to introduce statistics by defining it and giving its scope, applications, characteristics, • functions and limitations

Objectives:


To enlighten the students with the term statistics by providing various definitions•

To decide weather statistics is a science or an art•

To elaborate the scope, applications, characteristics, functions and limitations of statistics•

To explore as to why there is distrust on statistics•

Learning Outcome

At the end of this chapter students will be conversant with the following

Scope, applications, characteristics, functions and limitations of statistics•

Methods of data collection•

Statistical method of data interpretation & classification•

Importance of statistical methods in project preparation at different levels of methodologies•


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10.1 IntroductionStatistics can be referred as a subject that deals with numerical facts and figures. It is the set of mathematical tools and techniques that are used to analyze data. The word statistics is said to have been derived from the German word Statistik meaning political science or from Old Italian word stato meaning state or from New Latin word status meaning of which is position or form of government or political state. Statistical analysis involves the process of collecting and analyzing data and then summarizing the data into a numerical form.

10.2DefinitionofStatisticsThe word statistics refers either to quantitative information or to a method of dealing with quantitative information. There are many definitions to the term statistics given by different authors which are as given below:

Prof.A.L.Bowley• defined statistics as “Numerical statement of facts in any department of enquiry placed in relation to each other.”Websterdefined• statistics as “The classified facts respecting the condition of the people in a state especially those facts which can be stated in numbers or in tables of numbers or in any tabular or classified arrangement.”

10.3 Scope and Applications of StatisticsStatistics is associated with almost all the sciences as well as social, economic and political activities. The applications of statistics are so numerous and it is of great use to human beings in many ways. Science has become so important today that hardly any science exists independent of this and hence the statement-“Science without Statistics bear no fruit; Statistics without Science has no root.”

Statistical data and statistical methods are helpful in proper understanding of the economic problems and help • in solving a variety of economic problems such as wages, prices, analysis of time series etc. Statistical methods help in formulating economic policies and in evaluating their effectStatistical methods are being widely used in all business and trade activities like production, financial analysis, • distribution, costing, market research, man power planning, business forecasting etc. Business executives and managers rely mainly on statistical techniques to study the need and desire of the consumers.In industry, statistics is widely used in ‘quality control’. To find whether the product is confirming to specifications • or not, statistical tools like inspection plans, control charts etc are of great useA government’s administrative system is fully dependent on production statistics, income statistics, labour • statistics, economic indices of cost, price etc. All the departments of a government depend upon statistics for efficient functioningIn biology, medicine and agriculture, statistical methods are applied in the study of growth of plant, movement • of fish population in the ocean, migration of birds, effect of newly invented medicines, theories of heredity, estimation of yield of crop, effect of fertilizer on yield, birth rate, death rate, population growth, growth of bacteria etc

10.4 Characteristics of StatisticsSome of its important characteristics are given below:•

Statistics are aggregates of facts �Statistics are numerically expressed �Statistics are affected to a marked extent by multiplicity of causes. �

Statistics are enumerated or estimated according to a reasonable standard of accuracy. • Statistics are collected for a predetermine purpose �Statistics are collected in a systemic manner �Statistics must be comparable to each other �

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10.5 Functions of StatisticsThe various functions of statistics are as given below:• It simplifies themass of data.• With the help of statistical methods, the complex data is simplified into diagrammatic and graphical representations, averages etc.Itpresentsthefactsinadefiniteform.• Facts that are expressed in numbers are more convincing than expressed in statements. Statistics helps to present the data or facts in precise and definite form for easy understanding.It helps in comparison of data of same kind.•Statistical methods are extremely helpful in formulating and testing hypothesis and developing new •theories.It helps to predict future trends and to estimate any value of the population from the sample chosen.•It helps in bringing out the hidden relations between variables.•Withthehelpofstatistics,decisionmakingprocessbecomeseasier.•

10.6 Limitations of StatisticsStatistics, inspite of being widely used in many fields and being involved in every sphere of human activity, • faces certain limitations which are as follows:Statistics does not deal with qualitative aspects like honesty, intelligence etc. It deals with only quantitative • data.It does not study individual facts because individual items taken separately do not form a statistical data. • Statistical methods can be applied only to the aggregate of facts.Statistical tools do not provide the best solution to problems under all circumstances. It is one of the methods • of studying a problem and it should be supplemented by some other methods.Statistical analysis is based on probability and not on certainty. So statistical results are not universally true and • they are true only on an average.Common man cannot handle statistics properly, only statisticians can handle statistics properly.• The most important limitation of statistics is that they are liable to be misused and misinterpreted. Increasing • misuse of statistics has led to increasing distrust in statistics.

10.7ClassificationClassification refers to grouping of data into homogeneous classes and categories. A group or a class category has to be determined on the basis of the nature of the data and the purpose for which it is going to be used.

10.8ObjectivesofClassification

To condense th• e mass of data: Statistical data collected during the course of an investigation is in the raw form. With raw data we can’t make any conclusion unless it is properly classified into small groups or classes.

To prepare the data for tabulation: Only classified data can be presented in the tabular form.•

To study the relationships: Relationship between the variables can be established only after the various •

characteristics of the data have been known, which is possible only through classification.

To facilitate comparison: Classification helps us to find conclusions based onthecomparison of variables.•


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10.9CharacteristicsofClassificationThe following are general guiding principles for a good classification.• Exhaustive: Classification must be exhaustive. i.e. each and every item in the data must belong to one of the • classes.Mutually exclusive: Each item of information should fit only in one class, i.e. overlapping of items is not • allowed.Suitability: The classification should conform to the object of inquiry. For example, if the study is regarding the • economic condition of workers then classification must not be done on the basis of their religion.Homogeneity: The items included in each class must be homogeneous; Else there should be further classification •

in to sub groups.Flexibility: A good classification should be flexible. It should be adjustable•

To the new and changed situations and conditions. • Stability: The basic principle of classification should be retained throughout.

10.10 Frequency DistributionA classification according to the number possessing the same value of the Variable is known as frequency distribution of the given raw data.

Tally Marks ( | ) : It facilitates counting the frequency of a value of a variate in a systematic manner. The distinct values of the variate are written down in ascending or descending order in a column. As we go through the given raw data, one by one a tally mark is inserted in each case against the respective value. It will be easy to count if tally marks are arranged in blocks of five i.e. every fifth tally mark is marked by a slanting line over the preceding four. For example for the value of variate 5 we can give tally marks as ||||, for the value of variable 13 we can give tally marks as|||| |||||||.

10.10.1 Discrete or Ungrouped Frequency DistributionThis type of distribution shown above is called Discrete or Ungrouped Frequency Distribution. The ungrouped frequency distribution is quite handy if the values of the variables are largely repeated otherwise there is hardly any condensation.

10.10.2 Continuous or Grouped Frequency DistributionIn this form of distribution the frequencies refer to groups of values. This becomes necessary in the case of some variables which can take any fractional value and in whose case an exact measurement is not possible. e.g. the height, Weight income ,etc.

10.10.3 Cumulative Frequency DistributionIn cumulative distribution, the cumulative frequencies (c.f.) are derived by successively adding the frequencies of the successive individual class intervals. The cumulative frequency of a given class can be represented by the total of all the previous class frequencies including the frequency of that class. There are two types of cumulative frequencies.

‘• less than’ type: It will represent the total frequency of all classes less than and equal to the class value to which it relates.‘• more than’ type: It will represent the total frequency of classes more than and equal to the class value to which it relates.

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SummaryStatistics helps in creating more efficiency in the decision making process• Statistics can be said as a collection of methods for planning experiments, obtaining data, and then organizing, • summarizing, presenting, analyzing, interpreting, and drawing conclusionsThe purpose of statistics is to obtain some overall understanding of group characteristics• It is important to know how to understand statistics so that improper judgments are not made•

ReferencesJ.K. Sharma• , Business Statistics, Statistics, Dorling Kindersley Pvt. Ltd. 4th Impression, 2009, p1-26 & 27-60.T.R.Jain,S.C.Agarwal• , Statistics for BBA, Statistics, VK Enterprises, 2009-10, p1-13, 14-24.J. Medhi• , Statistical Methods, Methods of Data Collection, New Age International Publishers, First Edition, reprint-2005, p8-12.S.P.Rajagopalan and R.Sattanathan• . Business Statistics and Operations Research.2008.Tata McGraw-Hill Education.First Reprint Edition.p1-6

Recommended ReadingsLeidn university• , http://www.math.leidenuniv.nl/~redig/lecturenotesstatistics.pdf, last accessed date: 14th oct 2010Star,• Statistics http://www.stat-help.com/intro.pdf , last accessed date: 14th oct 2010Richald,• Statistics, http://people.richlan,.edu/james/lecture/m170, last accessed date: 14th oct 2010Artofproblemsolving,• Statistics, http://www.artofproblemsolving.com/LaTeX/Examples/statistics_firstfive.pdf last accessed date: 14th oct 2010Answers.com• , Statistics, http://www.answers.com/topic/statistics last accessed date: 14th oct 2010


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Self AssessmentStatistics deals with following1.

Qualitativea. Qualitative & Quantitativeb. Quantitativec. Productived.

Which of the following are the functions of statistics?2. It simplifies the mass of data.a. It presents the facts in a definite form.b. Statistics are numerically expressed.c. It gives us numerical representationd.

Which of the following is not a characteristic of Statistics?3. Statistics are aggregates of facts.a. Statistics are affected to a marked extent by multiplicity of causes.b. It helps in bringing out the hidden relations between variables.c. Statistics are collected in a systemic manner.d.

Common man cannot handle statistics properly, only _________ can handle statistics properly.4. techniciansa. statisticiansb. artisansc. administratorsd.

“By statistics we mean quantitative data affected to a market extent by multiplicity of causes.” This definition 5. is defined by:

Yule and Kendalla. Websterb. Prof.A.L.Bowleyc. Tippetd.

Statistics is an art or a science?6. Arta. Scienceb. Only a.c. Both a. and b.d.

Statistics can be referred as a subject that deals with ________ facts and figures.7. Alphanumericala. Alphabeticalb. Numericalc. Quantatitive d.

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“Statistics is both science and art. It is a science, as its methods are basically systematic and have general 8. applications. It is an art, as its successful application depends to a considerable degree on the skill and special experience of a statistician.” This statement is given by ___________.

Tippeta. Peter Druckerb. Webstarc. Oxfordd.

In industry, statistics is widely used in ___________.9. manufacturinga. salesb. quality controlc. distributedd.

______________of data leads to false conclusions.10. Misinterpretationa. Understandingb. Conceptualizationc. Summarizingd.


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Application I

Interpolation:In financial analysis • Interpolation is used widely in;Determination of IRR• Finding out yield to maturity bond debenture• Time value of money• Present & future value tables.• Example:• The cash inflows of a project involving an initial outlay of Rs.22 lakh is as follows;•

Year Rs. In lakh

1 10

2 10

3 6

4 3

The IRR is the rate at which the total value of discounted cash outflow is exactly equal to the disc. Cash inflows. The IRR of the project can be determined only through a process of trial and error.To begin with, let us try Disc rate of 14 %.Using PV interest factor table , the discounted cash inflows will be,(10x0.877) + (10x0.769) + (6x0.675) + (3x0.592)= Rs. 22.29 lakhSince it is higher than initial outflow, we must discount at higher rate.Likewise at 15 % interest rate(10x0.870)+(10x0.756)+(6x0.658)+(3x0.572) = Rs.21.93 LakhNow we know that as the rate increased by 1 % the Discounte cash flows falls from 22.29 to 21.93

Therefore the descent rate would be, around 22 lakh because it should be higher than 21.93 & lower than 22.29 lakhSo, The IRR = 14% + x 1 = 14 + 0.806

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Application II

Statistics:Statistics in School:• May be used to see the performance of students collectively in studies, sports or other activities• Gives information about the school’s population change every year• Helps in processing certain evaluations and surveys given to help improve the planning and co-ordination of • all school activitiesDecide the relationship of educational presentation to other factors like socioeconomic background•

Statistics in Social Science:Helps in providing the government more information about its citizens• Statistical results may initiate social reforms that would help benefit the standard of living•


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Application III

Statistics in SportsGivesabriefsummaryoftheeventsinagamewithhelpoftables,graphsandotherparameters

Statistics in Science• Rare species of different wildlife could be protected through regulations and laws developed using statistics• Epidemics and diseases are examined with the help of statistics• Helps in the evaluation of certain medical practices and the effectiveness of drugs•

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Solved Examples

Chapter IMatrices & Determinants

Q.1. Solve, x + y + z = 6

y + z = 3y – z = 1

Solution: Using Crammer’s Rule,

=

= 1(-1-1)-1(0)+1(0) = -2+0+0 = -2

== 6(-1-1)-1(-3-1)+1(3-1)

= -12+4+2

= -6

= = 1(-3-1)-6(0)+1(0)= -4+0+0 = -4

= = 1(1-3)-1(0)+6(0) = -2+0+0 = -2

Therefore, solution is as X = = -6/-2= 3Y = /= -4/-2= 2Z = = -2/-2= 1Therefore, X =3, Y = 2, Z = 1


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Chapter IIMathematical LogicQ. Solve according to De Morgan’s Law. Prepare truth table for, ~ (p∨ q) = ~p ∧ ~q Solution:

p q ~p ~q

T T T F F F FT F F T F T TF T F T T F TF F F T T T T

Q. Prove that, p→q = ~p⋁q = ~q→~pSolution:

p q p→q ~p ~q ~p⋁q ~q→~p

T T T F F T TT F F F T F FF T T T F T TF F T T T T T

Chapter IIISet Theory

Q. In a class of 42 students, every student studies at least one of the subjects. Mathematics (A), English (B) and Commerce (C). 14 students study Mathematics, 20 Commerce and 24 English. 3 students study Mathematics and Commerce, 2 study English and Commerce and there are no students who study all the three subjects. Find the number of students who study Mathematics and English and not Commerce.

Solution:Given that: U=42, n(A)=14, n(B)=24, n(C)=20, n(A∩C)=3, n(B∩C)=2, n(A∩B∩C)=0

We have to find n(A∩B)=?n(A∪B∪C) = U = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)42 = 14+24+20-n(A∩B)3-2+0

Therefore, solving n(A∩B)=11Number of students who study Mathematics and English and not Commerce = 11

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Q. It is known that in a group of people, each of the members speaks at least one of the languages English, German, and Russian. 31 speaks English, 36 speaks German, 27 speaks Russian, 10 speak both English and German, 9 both English and Russian and 11 both German and Russian. Prove that the group contains at least 64 people and not more than 73 people.

Let A represent English, B represent German and C represent Russian.

Given that : n(A)=31, n(B)=36, n(C)=27,n(A∩B)=10,n(A∩C)=9, n(B∩C)=11

n(A∪B∪C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C) = 31+36+27-10-11-9+n(A∩B∩C)

n(A∪B∪C) = 64+n(A∩B∩C)If there is no person who speaks all the three languages then n(A∩B∩C)=0Thus people in the group =64+0 =64. But, if there are some persons who speak all the three languages then, Considering, n(A∩B)=10,n(A∩C)=9,n(B∩C)=11We have, n(A∩B∩C)=9(cannot be more than9)

Thus the Maximum number of people in the group = 64+9 = 73.

Chapter VIArithmetic & Geometric Progression

Q. A gas lighter has to light 100 gas lamps. He takes 1.5 minutes to go from one lamp post to the next. Each lamp burns 10 cubic feet of gas per hour. How many cubic feet of gas has been burnt by 8.30 p.m., if he lights the first lamp at 6.00 pm.?

Solution:

First lamp burns for 150 minutes. (From 6.00 to 8.30pm.)Second lamp burns for, (150 - 1.5)Third lamp burns for, (150 - 2 x 1.5)Fourth lamp burns for, (150 - 3 x 1.5)Similarly, 100 lamp burns for (150 - 99 x 1.5)Therefore, Total time = 150 x 100 - 1.5(1+2+3+…….+99)

= 15000 - 1.5 (99 x 100 ) Since, 1+2+3….+99 in A.P.2= 15000 -7425 = 7575 minutes.

Therefore, Total gas burnt = 7575 (10/60)

= 1262.50 cubic feet.


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Q. A car is purchased for Rs. 80,000. Depreciation is calculated at 5 % per annum for the first 3 years and after that at 10 % per annum for the next 3 years; depreciation is calculated on the diminishing value. Find the value of the car after a period of 6 years.Solution:Let the cost of the car = Rs. PDepreciation in the first year = P x 5/100………… (5%)Depreciated value after the first year = cost depreciation= P - (P x 5/100)= P (1- (5/100))Depreciation for the second year = P (1- (5/100)) x 5/100, Since (5% on diminishing value)Depreciated value after the second year is as,= [P (1- (5/100))] - [P (1- (5/100)) x 5/100 ], Can be simplified as,= P [1- (5/100)]2

Similarly after 3 years = P [1- (5/100)]3 , For the next 3 years 10 % depreciation is [1-(10/100)]3

Therefore, after 6 years the value of the car is, = P [1 - (5/100)]3 x [1-(10/100)]3

(Since P = cost of the car = 80,000 given)= 80,000 x [1 - (5/100)]3

x [1 - (10/100)]3

= 50,002

Chapter VProbabilityQ. An urn contains 3 white and 5 red balls and another urn contains 2 white and 4 red balls. One urn is selected at random and a ball is drawn from it at random.Find the probability that the ball drawn is red.

Solution:The required event happens if one of the following two mutually exclusive events happens.

Let event A : First urn is selected and the ball drawn is red.Let event B : second urn is selected and the ball drawn is red.

Probability that out of two urns, the first one is selected is ½.When it is known to be selected, probability that the ball is red for the first urn is 5/8.

Thus, P(A) = (½)(5/8) = 5/16When it is known to be selected, the probability that the ball is red for the second Urn is 4/6Thus P(B) = (½) (4/6) = 1/3Required probability = P(A∪B) = P(A) + P(B)

= 5/16 + 1/3 P = 31/48.

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Q. If P(A) = ½, P(B) = 1/3, P(A∩B) = ¼, Find P(A/B), P(A∪B), P(A‘∩Β’).Solution:P(A/B) = P(A∩B)/ P(B) = (¼)/1/3) = ¾P(A∪B) = P(A) + P(B)- P(A∩B) = ½ + 1/3 ¼ = 7/12P(A‘∩Β’) = P(A ∪ B)' ……… Using De Morgan's Law

= 1 - P(A∪B) . …….. [ P(A) + P(A') = 1 ]= 1 - 7/12= 5/12.

Chapter VIPermutations & CombinationsQ. 4 men and 3 women are to be seated for a dinner such that no 2 women sit together and no 2 men sit together. Find the number of ways in which this can be arranged?Solution:Let the seven chairs be numbered as 1, 2, 3, 4, 5, 6, 7Now 3 women can be placed in 3 even places in 3P3 ways = 3! In the remaining 4 places 4 men can be arranged as 4P4 ways = 4!

The total number of ways of arrangement according to the given condition is3! X 4! = 144.

Q. Find n and r, if nPr = 110, nCr = 55.

Solution:We know that, nCr =

55 =

Therefore, by solving, r! = 2Since, r! = 2 gives r = 2nP2 = n(n-1) = 110by solving , we get n = -10 or n = 11But n can’t be negative. Thus,n=11


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Chapter VIIICommercial ArithmeticQ. A salesman receives (9/2) % commission on sales upto Rs.3000 and 5% commission on the sales over Rs.3000. Find his total income on a sale of Rs. 7500.

Solution:Since, on Rs.100 the commission is 9/2.

Therefore, on Rs. 3000 the commission is = 135

Thus,forsalesupoRs.3000commissionis135

Now sales over Rs. 3000 is (7500-3000) = 4500Since on Rs.100 the commission is Rs. 5Therefore, on Rs. 4500 the commission is 4500x 5 = 225100Thus,salesoverRs.3000,commissionis225Therefore, the total commission on a sale of Rs. 7500 = 135+225 = Rs. 360

Q. Find the present worth of Rs. 560 due 3 years hence at 4% per annum simple interest. Find also the true discount.Solution: Let Present Worth (P.W.) = Rs. PS.D. = P.W. + T.D.S.D. = P + P.n.r /100 ………………simple interest = pnr/100Where , n= no of yrs. & r = rate of interest.S.D. = P ( 1 + n.r/100) …………….equation (1)Let n = number of years = 3r = rate of interest = 4S.D.= 560 (given)Thus, solving equation (1), we get P = 500T.D. = S.D - P.W.T.D. = 560 - 500 = 60Thus,Present Worth = Rs.500True Discount = Rs. 60

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Chapter IXRelation & FunctionsQ. A company has fixed costs of Rs. 26,000. The cost of producing one item is Rs. 30. If this item sells at Rs. 43, what is the break-even point?

Solution:Let x = number of items produced and soldC(x) = Total cost = fixed cost + variable cost= 26,000 + 30x

R(x) = Total revenue = 43x

At the break-even point, C(x) = R(x)26,000 + 30x = 43x

By solving, x = 2,000 items.Therefore,break-evenpointisat2,000items.


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ReferencesRajendra Akerkar, Discrete Mathematics: Set theory, Dorling Kindersley Publication India. 2008. 2nd Impression • 2009. P109-123.T. Veeraranjan, Discrete Mathematics with graph theory & Combinatorics: Set theory, McGraw-Hill Publication, • 7th Edition. 2008. P51-64.Seymour Lipschutz, Set theory and related topics: Set Theory, Mcgraw-Hill Publication. 1998, 2nd Edition. • P1-35.H. Dubner, Large Sophie Germain primes, Math. Comp. 65 (1996), 393-396. • R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer-Verlag, 1994, pp. 15-18. • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, New • York, 1979. L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp. 21 (1967), 489. • H. L. Nelson, There is a better sequence, J. Recreational Math. 8 (1) (1975), 39-43. • P. A. Pritchard, A. Moran, and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. 64 • (1995), 1337-1339. Charles Miller Grinstead, James Laurie Snell, Introduction of Probability: Probability, AMS Bookstore, 1997, • p133-137Frederick Mosteller, Probability: Probability, Dover Publications, 1987, First Edition, p1-100• Dartmouth, Probability http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/• book.pdf, last accessed: 18 oct 2010Math Goodies, Probability, http://www.mathgoodies.com/lessons/vol6/intro_probability.html, last accessed: • 18 oct 2010Britannica Encyclopedia, Probability, http://www.britannica.com/EBchecked/topic/477530/probability-theory, • last accessed: 18 oct 2010Krishna R. Kumar, Discrete Mathematics, Permutation & Combination, Firewall Media, 2005, p23-47.• G. Shankar Rao, Discrete mathematical structure, Permutation, New Age International, 2002, p14-54.• R.C. Pennar, Discrete Mathematics, Combination, World Scientific, 1999, p54-67.• TR Jain, AS Sandhu, Quantitative Methods: Interpolation, VK Publication, 2006-07, p7.1-7.38• B.R.Agarwal, Programmed Statistics, Interpolation, New Age International, 2nd Edition, 2007, p405-425.• N.P Bali, P.N.Gupta, A textbook of Quantitative Techniques, Interpolation, Laxmi Publications, 1st edition, • 2008, p134-145.G.R. Veena, Business Mathematics, Commercial Arithmetic, New Age International Publishers, 2006, p215-• 255, 268-270R.S Aggarwal, Quantitative Methods, S.Chand Publications, 2008, p251-293 & 632-636• Oscar Williams, Commercial Arithmetic, Profit & Loss Bibliobazar, 2008, p3-85• S. Jackson, Commercial Arithmetic, Special Application, 2009, p23, 45-67.• Geo H. Douglas, Modern Commercial Arithmetic, Business Mathematics, Commision agent, 2007, p34-90• TR. Jain, Quantitative Methods, Relation & Functions, FK Publication, 2nd Edition, p54• Author Stream, Relation & Functions, www.authorstream.com/.../sadamava-373982-2-1-relations-functions-• ppt-relationsfunctions-powerpoint-education/, Last accessed-10th October 2010.Tutor Vista.Com, Relation & Functions, www.tutorvista.com › Math › Algebra, Last accessed-10th October • 2010.

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Purple Math, Relation & Fuinctions, www.purplemath.com/modules/fcns.htm, last accesed-10th Ocober • 2010.J.K. Sharma, Business Statistics, Statistics, Dorling Kindersley Pvt. Ltd. 4th Impression, 2009, p1-26 & 27-• 60.T.R. Jain, S.C.Agarwal, Statistics for BBA, Statistics, VK Enterprises, 2009-10, p1-13, 14-24.• J. Medhi, Statistical Methods, Methods of Data Collection, New Age International Publishers, First Edition, • reprint-2005, p8-12.S.P.Rajagopalan and R.Sattanathan. Business Statistics and Operations Research.2008.Tata McGraw-Hill • Education.First Reprint Edition.p1-6

Recommended ReadingDr. V.N. Kala, Rajashri Rana. Matrices. 2009.Laxmi Publication ltd. First Edition. P1-25 & 140-144.• TR Jain, SC Agarwal. Business Mathematics & Statistics. V.K Enterprises.2009-10. Revised. P1-40.• TR Jain, SC Aggarwal, Quantitative Methods. 2008-09. FK Publication. P1-92.• J.O.Bird(2001), Newnes engineering mathematics pocket book, Matrices & determinants, Butterworth & • hannmen, p230-240.Gareth Williams (2009), Gareth Williams, Matrices & determinants, Jones& Bartlett Publications, 7th Edition, • p69-165. David Poole(2005), Linear Algebra, Matrix, Cengage learning, 2nd Edition, p134-150.• David McMahon, Linear Algebra Demystified, Matrices, McGraw-hill publication, 2005, p34-74• Howard Anton, Elementary Linear Algebra, Matrices, FM Publications, 10th edition 2010, p1-106.• Warner Greub, Linear Algebra graduate texts in mathematics, Springer, 1975, p83-131 • Leidn university, http://www.math.leidenuniv.nl/~redig/lecturenotesstatistics.pdf, last accessed date: 14th oct • 2010Star, Statistics http://www.stat-help.com/intro.pdf , last accessed date: 14th oct 2010• Richald, Statistics, http://people.richlan,.edu/james/lecture/m170, last accessed date: 14th oct 2010• Art of problem solving, Statistics, http://www.artofproblemsolving.com/LaTeX/Examples/statistics_firstfive.• pdf last accessed date: 14th oct 2010Answers.com, Statistics, http://www.answers.com/topic/statistics last accessed date: 14th oct 2010• Joseph S. Fulda (1993). Exclusive Disjunction and the Bi-conditional: An Even-Odd Relationship. Mathematics • Magazine 66 (2):124.Philip P. Hallie (1954). A Note on Logical Connectives. Mind 63 (250):242-245.• Dean P. McCullough (1971). Logical Connectives for Intuitionist Propositional Logic. Journal of Symbolic • Logic 36 (1):15-20.Heinrich Wansing (2006). Logical Connectives for Constructive Modal Logic. Synthese 150 (3)• Donald Waters(2006), Quantitative Methods for business, Set Theory, Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Set theory, Elsevier, • J. Curwin Slater(2007), Quantitaitve Methods, Set theory, Thomson Learning, • Donald Waters(2006), Quantitative Methods for business, Progression, Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Arithmetic Progression, Elsevier• J. Curwin Slater(2007), Quantitaitve Methods, Progression, Thomson Learning• Donald Waters(2006), Quantitative Methods for business, Probability , Prentice Hall Publication,4th Edition• Diana Bedward (1999), Quantitative methods, Probability, Elsevier • J. Curwin Slater(2007), Quantitaitve Methods, Probability , Thomson Learning,•


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Donald Waters(2006), Quantitative Methods for business, Permutation & Combination, Prentice Hall • Publication,4th EditionDiana Bedward (1999), Quantitative methods, Permutation, Elsevier,.• J. Curwin Slater(2007), Quantitative Methods, Combination ,Thomson Learning.• Waters Donald(2006), Quantitative Methods for business, Interpolation, Prentice Hall Publication,4th • EditionBedward Diana(1999), Quantitative methods, Interpolation, Elsevier, • J. Curwin & Slater(2007), Quantitaitve Methods, Interpolation of Polynomials, Thomson Learning,• John Henry Moore, New Commercial Arithmetic, Bibliobazaar LLC, 2008, p50-145.• Augustus D morgan, Elkements of Arithmetic, Taylor & Walton, 1900, p125-150.• Frederick calder, Elementary rules of Arithmetic, revised edition, 1852, p106, 171.• Donald Waters(2006), Quantitative Methods for business, Relation & fuinction, Prentice Hall Publication,4th • EditionDiana Bedward (1999), Quantitative methods, Relation & Functions, Elsevier.• J. Curwin Slater(2007), Quantitaitve Methods, Relation & Functions, Thomson Learning.• Leidn university, http://www.math.leidenuniv.nl/~redig/lecturenotesstatistics.pdf, last accessed date: 14th oct • 2010Star, Statistics http://www.stat-help.com/intro.pdf , last accessed date: 14th oct 2010• Richald, Statistics, http://people.richlan,.edu/james/lecture/m170, last accessed date: 14th oct 2010• Art of problem solving, Statistics, http://www.artofproblemsolving.com/LaTeX/Examples/statistics_firstfive.• pdf last accessed date: 14th oct 2010Answers.com, Statistics, http://www.answers.com/topic/statistics last accessed date: 14th oct 2010•

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Self Assessment Answers

Chapter I1. c2. a3. d4. b5. b6. a7. a8. a9. a10. a

Chapter II1. a 2. d 3. c 4. b 5. b 6. d 7. a 8. a 9. P Q ~P ~QT T F FT F F TF T T FF F T T

10.P Q ~P ~Q ~P v ~Q P ^ (~P v ~Q)T T F F F FT F F T T TF T T F T FF F T T T F

Chapter III6. False7. True8. False9. True10. a) There are two sets in three elements, {1, 2, 3} & {6, 7, 8}. Hence B= [{1, 2,3},{6,7,8}] b) There are no sets in A with four elements; hence B is empty, that is, B=∮.11. -{a,b,c,d,e,f,g} 12. -{b,g} 13. -{e,g,h}14. -{d,e}15. . {7, 11, 16}16. . {7,11,13,16,17}


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Chapter IV1. b 2. a 3. d 4. b 5. c 6. b 7. a 8. a 9. d 10. b

Chapter V1. b 2. b 3. d 4. a 5. b 6. d 7. c 8. a 9. b 10. b

Chapter VI1. b 2. b 3. b 4. b 5. b 6. c 7. a 8. d 9. b 10. c

Chapter VI1. a 2. b 3. a 4. d 5. c 6. c 7. a 8. d 9. d 10. b

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Chapter VIII1. a2. c3. b4. d5. a6. b7. b8. b9. a10. b

Chapter IX1. domain = -1.2.1.8.9 range= 2.51.3.222. domain= -5.21.11.81.19 range= 6.-51.93.202.513. Relation #1 and Relation #3 are both functions.4. Relation #1 and Relation #3 are functions because each x value, each element in the domain, has one and only one y value, or one and only number in the range. 5. Relation #2 is only function.6. a7. a8. d9. a10. d

Chapter X1. b 2. d 3. c 4. b 5. a 6. d 7. c 8. a 9. c 10. a

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Quantitative Techniques in Businessjnujprdistance.com/assets/lms/LMS JNU/BBA/Sem III...With the help of matrices & determinants, we can solve equations & system of equations. 1.2 Matrix