Business Mathematics II Lecture 2

8/18/2019 Business Mathematics II Lecture 2

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Prichestt and Saber

Bowen Mathematics

Appendix 2

Set Theory


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Set Specification

Roster Method

We can denote a set S in writing by listing all of its elements incurly braces:

{a, b, c} is the set of whatever 3 objects are denoted by a, b, c.

Descriptive Method

State the rules or conditions that distinguishes members of the setfrom non-members

For any proposition P (x) over any universe of discourse, {x| P (x)} is the set of

all x such that P(x).e.g., {x | x is an integer where x>0 and x


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Basic properties of sets

Sets are inherently unordered :

No matter what objects a, b, and c denote,{a, b, c} = {a, c, b} = {b, a, c} ={b, c, a} = {c, a, b} = {c, b, a}.

All elements are distinct (unequal);multiple listings make no difference!

{a, b, c} = {a, a, b, a, b, c, c, c, c}. This set contains at most 3 elements!


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Definition of Set Equality

Two sets are declared to be equal if and only if theycontain exactly the same elements.

In particular, it does not matter how the set is defined ordenoted.

For example: The set {1, 2, 3, 4} ={x | x is an integer where x>0 and x0 and


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Infinite Sets

Conceptually, sets may be infinite (i.e., not finite, withoutend, unending).

Symbols for some special infinite sets:N = {0, 1, 2, …} The natural numbers.Z = {…, -2, -1, 0, 1, 2, …} The integers.

R = The “real” numbers, such as

374.1828471929498181917281943125… Infinite sets come in different sizes!


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The Empty Set

Φ (“null”, “the empty set”) is the unique set that

contains no elements whatsoever.

Φ = {} = { x| lse}


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Venn Diagram

The Venn Diagram is made up of two or more overlapping circles.It is often used in mathematics to show relationships between sets.In language arts instruction, Venn Diagrams are useful forexamining similarities and differences in characters, stories,

poems, etc.

http://upload.wikimedia.org/wikipedia/en/0/06/Venn-diagram-AB.svg


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Types of Sets

Subset

Proper subset

Improper subset

Universal set (Ω

) Complement


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Set Operations

Unions Intersection

Disjoint


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Disjointedness

Two sets A, B are calleddisjoint (i.e., unjoined)iff their intersection isempty.

Example: the set of evenintegers is disjoint withthe set of odd integers.

Help, I’ve

been

disjointed!


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Set Symbols

byimpliedisandImlies

sec

)(

tion Inter Union

Subset

Subset Subset Not

set proper Subset

Element Not Element


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Null Set

Smallest subset of a super set is a se that contains no elementin common at all. It is called null set or empty set Ø or { }

Prove:

Null set is unique.

There is only one such set in the whole world and isconsidered a subset of any set that can be conceived.

ShenceS,say thatnotcanWe

}{,

}{,

However

S xand xthenS If


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Set Equality

Two sets are equal if they contain exactly the same elements,and we write, X = Y

Y X X Y and Y X


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Set Difference

For sets A, B, the difference of A and B, written AB, is the setof all elements that are in A but not B.

A B : x x A xB x x A xB

Also called:The complement of B with respect to A.


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Union

The union of two sets A and B is the set of elements in one orother of the sets.

}:{ Bor X x xY X C


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Set Difference Examples

{1,2,3,4,5,6} {2,3,5,7,9,11} = ___________

Z N {… , -1, 0, 1, 2,… } {0, 1,… }= {x | x is an integer but not a nat. #}

= {x | x is a negative integer}= {… , -3, -2, -1}


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Set Identities

Identity: A= A AU = A

Domination: AU=U A=

Idempotent: A A = A = A A

Double complement: Commutative: AB=B A AB=B A

Associative: A(BC )=( AB)C A(BC )=( AB)C

A A )(


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Problem

A company studies the product preferences of 20000customers. It was found that each of the products A, B, C wasliked by 7020, 6230, and 5980 respectively and all productswere liked by 1500; products A and B were liked by 2580,

product A and C liked by 1200 and products B and C liked by1950. Prove that the study results are not correct.


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Documents

Business Mathematics II Lecture 2