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Mathematics Division,IMSP,UPLB 2

AXIOMS

(or postulates)

THEOREMS

(proposition, lemma, theorem, corollary

which are deductively proven)

Primitive/undefined

terms

Definitions


Operations


Basic Operations 1. Addition

- denoted by +

- result is called sum

Ex. 2 + 3 = 5

Recall the rules for adding signed numbers

2 + 7 = _____ (-3) + (-5) = _____

8 + (-4) = _____ 4 + (-8) = _____


2. Multiplication

- denoted by x or , or just by juxtaposition

- result is called product

Ex. 2 x 3 = 6

Basic Operations

Recall the rules for multiplying signed numbers

2 x 8 = ____ (-3) x 6 = _____

3 x (-6) = ____ (-4) x (-5) = _____


Closure Property

A set is closed (under an operation) if and only if the operation on two elements of the set produces another element of the set.

If an element outside the set is produced, then the operation is not closed.

“When you combine any two elements of the set, the result is also

included in the set.”


Example If you add two real numbers, you will get

another real number.

Example: 2.5 + 4 = 6.5

Since this process is always true, it is said that

the the set of real numbers is

“closed under the operation of addition”.

“There is simply no way to escape the set of

real numbers when performing addition.”


Is the set of real numbers closed under multiplication?

If you multiply two real numbers, you will

get another ____ _______.

Example: 2(1/5)=2/5

Since this process is always true, it is said that the

the set of real numbers is

“closed under the operation of ___________”.

“There is simply no way to escape the set of

real numbers when multiplying.”

real number

multiplication


Time to think Consider the set E of even numbers.

1. Is E closed under addition?

2. Is E closed under multiplication?

2m+2n=2(m+n) (2m)(2n)=2(2mn)

Consider the set O of odd numbers.

1. Is O closed under addition?

2. Is O closed under multiplication?

3+3=6 (2m+1)(2n+1)=?


Time to think

Consider the set Q of rational numbers.

1. Is Q closed under addition?

2. Is Q closed under multiplication?

bd

cbad

d

c

b

a

integer

integer

bd

ac

d

c

b

a

integer

integer


Time to think

Consider the set Qc of irrational numbers.

1. Is Qc closed under addition?

2. Is Qc closed under multiplication?

222

022


Time to think Is P (set of prime numbers) closed under addition?

multiplication?

Consider 2+2 and 2(2)

Is C (set of composite numbers) closed under

multiplication? OBVIOUS!

RECALL Subtraction and Division:

Is N closed under subtraction?

Is Z closed under division?


Note: We can define subtraction in terms of addition.

Example:

1)3(232

1)2(323

BUT WE NEED TO SATISFY FIRST AN IMPORTANT

PROPERTY: EXISTENCE OF ADDITIVE INVERSE…


Note: We can define division in terms of

multiplication. But remember, we cannot divide

by zero!

Examples:

3

2

3

1232

2

3

2

1323

BUT WE NEED TO SATISFY FIRST AN IMPORTANT

PROPERTY: EXISTENCE OF MULTIPLICATIVE INVERSE…

Mathematics Division,IMSP,UPLB

15

Binary Operation Motivation: We can define our own operation.

A (closed) binary operation * is a type of operation

where there are two operands such that

1) The operands come from the same (non-empty) set

2) The set where the operands came from is closed

under the operator

3) The result of the operation is unique

x * y = z

Operands Operator

Result

Mathematics Division,IMSP,UPLB

16

Binary Operation Examples

+2 is a binary operation on Z2={0,1}

Z2 x Z2 Z2

(0,0)

(1,0)

(0,1)

(1,1)

0

1

1

0

0 +2 0 = 0

1 +2 0 = 1

0 +2 1 = 1

1 +2 1 = 0

Binary Operation Examples

“Followed by” is a binary operation on DC.

The usual + and x are binary operations on R.

Subtraction is not a binary operation on N but a binary

relation on _____?

Division is not a binary operation on R but a binary

operation on R – {0} .

Exponentiation is a binary operation on N but not on R,

why?

Binary Operation on R

The convention is to operate from left to right.

3+2+6+0+1

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

= (5+6)+0+1

= (11+0)+1

= 11+1

= 12.

We usually follow the order of operation: PEMA

(parentheses, exponents, multiplications, additions).


Properties of Real

Numbers


In this lesson we look at some

properties that apply to all real

numbers. If you learn these

properties, they will help you solve

problems in algebra. Let's look at

each property in detail, and apply it

to an algebraic expression.


Closure (Addition and Multiplication)

Associativity (Addition and Multiplication)

Existence of Identity (Addition and

Multiplication)

Existence of Inverses (Addition and

Multiplication)

Commutativity (Addition and Multiplication)

Distributivity (Multiplication over Addition)

Why this set of

axioms are

called Field

Axioms?


1. Commutative property

a) Addition: For all real numbers a, b

a + b = b + a

-we can add numbers in any order

b) Multiplication: For all real numbers a, b

a x b = b x a

-we can multiply numbers in any order


Time to think

a.) Is subtraction commutative in R?

b.) Is division commutative in R?


2. Associative property

a) Addition: For all real numbers a, b, c,

a + (b + c) = (a + b) + c

we can cluster numbers in a sum in any way we want and still get the same answer

b) Multiplication: For all real numbers a, b, c

(a x b) x c = a x (b x c)

we can cluster numbers in a product in any way we want and still get the same answer


Time to think

a.) Is subtraction associative in R?

b.) Is division associative in R?


3. Distributive property of multiplication over addition

For all real numbers a, b, c

a(b + c) = ab + ac

(left-hand distributive law)

and

(a + b)c = ac + bc

(right-hand distributive law)


4. Existence of Unique Identity Element

a) Addition: There exists a real number 0 such

that for every real a, a + 0 = 0 + a = a

- Zero added to any number is the number

itself.

- 0 is called the “additive identity”

b) Multiplication: There exists a real number 1

such that for every real a, a x 1 = 1 x a = a

- Any number multiplied by 1 gives the

number itself.

- 1 is called the “multiplicative identity”


5. Existence of Inverses

a) Additive Inverse (Opposite Sign)

For every real number a there exists a real number, denoted (-a), such that

a + (-a) = (-a) + a = 0

b) Multiplicative Inverse (Reciprocal)

For every real number a except 0 there exists a real number, denoted by 1/ a, such that

a x (1/ a) = (1/a) x a = 1

It is important to first have an identity element before having

inverses…

Examples

1. What is the additive inverse of 0?

0 0 0 by Existence of

Additive Identity

0 0

2. What is the additive inverse of ?a


(not part of the field axioms)

We can always find another real number

that lies between any two real numbers.


Real numbers satisfy the field axioms.

We need to be familiar with the properties of R in order to solve algebra problems.

Do you think we can solve algebra problems without these properties?

SUMMARY


Exercises

1.Name all subsets of R to which these numbers

belong:

a. 12 d. 2.3434… g. /3

b. –23 e. 27/3 h. e

c. 0 f. –3.45

2. Estimate the position of the numbers in #1 on the

number line.


Exercises

3. Let U=R

a) W Z d) Z+ N g) W - N

b) Z R e) Wc Z h) Rc

c) N Q f) Z – N i) Qc - Q


4) State the property of R that justifies the truth of the

following statements:

a) f) (e + ) + 2 = e + ( + 2)

b) g) There are infinitely many

real numbers between

c) 234 + 345 = 345 + 234 2 and 2.1

d) (2+1)+5=5+(2+1) h) 23 + (-23) = 0

e) 2(3+2)=6+4 i) 7(1/7) = 1

37 3 R

2 35

3 23R

Exercises

Answer is Commutativity not Associativity!


Reflections 1. What is a number? How is number useful

to our daily lives?

2. Name the subsets of the set of real

numbers.

3. What are the “nice” properties that the set

of real numbers obey under the

operations of addition and multiplication?

4. Why do we consider these properties as

“nice”?


Equality Axioms

Equality Axioms

Property of Equality

For any

R

eflexive

, .a a aR


For , ,

Symmetric

if then .a bR bb aa


For , , , if and

then .

Transitive

aa b c R b b c

a c

Equality Axioms

Addition Property of Equality APE

For , , , if then a bR aa cc bb c

Multiplication Property of Equality MPE

For , , , if then a b a ca b R bc c

Substitution

If two real numbers are equal, then one

may be substituted for the other in any

algebraic expression.

If 5 , thenx y 2 3 2 5 3 .x y y y

33Also, 5 .x y

Solvable equations

and Groups


Solvable Equation

Consider 3 5.x

2 is a solution and 2 N.x

The equation is solvable in N.

Now consider 3 0.x

This equation is not solvable in N.

: N is not large enough to contain

the solutions even for such simple linear

equations that we

Somethin

h

g

ave

must

Re

b

seen

e do

mark

.

ne!

Solvable Equation

Is 3 3 solvable in ?x N

Is 3 2 solvable in ?x N in ?W

in W?

in Z?

FYI: [Z,+] is a group.

But what is a group?

What properties of reals are

needed to solve any linear

equation of the form a+x=b?

- Closure (under +)

- Associativity (for +)

- Existence of additive identity

- Existence of additive inverse

a+x = b

(a+x)+(-a) = b+(-a) APE

(a+(-a))+x = b+(-a) Associativity (+)

0+x = b+(-a) Exist. of Add. Inverse

x = b+(-a) Exist. of Add. Identity

Group (an algebraic structure)

Given a non-empty set and an operation

on , the mathematical system

is a

,

gro ifup

G

G

G

closed

associative

identity

1. is under .

2. is in .

3. There is an element under in .

4. Every element of

inver sehas an under .

G

G

G

G

* is a binary

operation on G

Examples

Which of the following mathematical

systems are groups?

1. ,N

Is closed under +?N

Is + associative in ?N

What is the identity element?

Therefore, , is not a group.N

Examples

2. ,Z

Is closed under ?Z

Is associative in ?Z

Counterexample

1 2 3 1 1 2

1 2 3

1 2 3 1

3 4

2

1

3

Therefore, , is not a group.Z

Examples

3. ,N

4. ,Q

5. , Z

6. [Q – { 0}, ∙]

TO DO

Determine the identity element

and inverse of each element of

[DC, “followed by”].

Any linear equation of the

form a*x=b is solvable if

a,bϵG and [G,*] is a group.

Abelian Group

A group is an if its

operation is

abelian group

commutative.

Example

, is a group.Z

Is commutative on ?Z

Therefore, , i abelian gs ran .oupZ

Solvable Equation

Is 3 6 solvable in ?x Z

Is 3 1 solvable in ?x Z

We need multiplicative inverses!

Solvable Equations

Is 2 3 0 solvable in ? in ? in ?x Z Q R

Remark: ax+b=0 is solvable in a field.

But before we discuss the concept of a field,

we will discuss first the concept of a ring.

Rings and Fields are also algebraic

structures.

We need multiplicative inverses

and we have TWO operations!

Documents

Ako, si , ay nangangakong magsisipag mag-aral hindi lang para sa aking …jfrabajante.weebly.com/uploads/1/1/5/5/11551779/5_math_17.pdf · Ang aking kontrata: Ako, si _____, ay nangangakong