Number Sets, Field Axioms and Order Axioms

8/9/2019 Number Sets, Field Axioms and Order Axioms

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Math 11: College Algebra


Reymart S. Lagunero

Department of Mathematics and Computer ScienceUniversity of the Philippines Baguio

February 2, 2015

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Number Sets

Number Sets

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Number Sets

The Algebra of Counting Numbers

The Real Number System

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Number Sets


Definition

An integer is a real number which is either a counting number, or0, or the negative counting numbers. That is,

Z = {x |x ∈ N ∨ x = 0 ∨ x = −n ∀n ∈ N}.

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Number Sets


Definition

An integer is a real number which is either a counting number, or0, or the negative counting numbers. That is,

Z = {x |x ∈ N ∨ x = 0 ∨ x = −n ∀n ∈ N}.

Definition

If n, m and k are integers and n ·m = k , then n and m are said tobe factors or divisors of k and k is said to be a multiple of n andm.

M 11 C A

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Number Sets


Definition

A positive integer is called composite if it is different from 1 andcan be expressed as the product of two or more positive integersdifferent from itself.

Ma 11 Co g A g a

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Number Sets


Definition

A positive integer is called composite if it is different from 1 andcan be expressed as the product of two or more positive integersdifferent from itself.A positive integer is called prime if it is different from 1 and notcomposite.


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Number Sets


Definition

A positive integer is called composite if it is different from 1 andcan be expressed as the product of two or more positive integersdifferent from itself.A positive integer is called prime if it is different from 1 and notcomposite.Two integers are relatively prime if they have no common prime

factors, so that the only factors they have in common are 1 and -1.


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Number Sets


Definition

A real number x which can be expressed as a quotient of an integern by a nonzero integer m is called a rational number. That is,

Q = {x |x = n

m ∀ 0 = m, n ∈ Z}.


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Number Sets


Definition

A real number x which can be expressed as a quotient of an integern by a nonzero integer m is called a rational number. That is,

Q = {x |x = n

m ∀ 0 = m, n ∈ Z}.

A real number which is not a rational is called an irrational

number denoted by Q

.


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Number Sets

Field Axioms for R

Axioms1A Closure for Addition: ∀a, b ∈ R, a + b ∈ R.

1M Closure for Multiplication: ∀a, b ∈ R, ab ∈ R.

2A Associativity for Addition:

∀a, b , c ∈ R, (a + b ) + c = a + (b + c ).2M Associativity for Multiplication: ∀a, b , c ∈ R, (ab )c = a(bc ).

3A Commutativity for Addition: ∀a, b ∈ R, a + b = b + a.

3M Commutativity for Multiplication: ∀a, b ∈ R, ab = ba.

4 Distributive Property of Multiplication Over Addition:∀a, b , c ∈ R, c · (a + b ) = c · a + c · b and(a + b ) · c = a · c + b · c .


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Number Sets

Field Axioms for R

Axioms

5A Existence of Additive Identity:∃! 0 ∈ R . a + 0 = 0 + a = a ∀a ∈ R.

5M Existence of Multiplicative Identity:∃! 1 ∈ R . a · 1 = 1 · a = a ∀a ∈ R.

6A Existence of Additive Inverse:∃! (−a) ∈ R . a + (−a) = (−a) + a = 0 ∀a ∈ R.

6M Existence of Multiplicative Inverse:∃! ( 1a

) ∈ R . a( 1a

) = ( 1a

)a = 1 ∀0 = a ∈ R.


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Number Sets

Field Axioms for R

Axioms

E1 Reflexivity for Equality: ∀a ∈ R, a = a.

E2 Symmetry for Equality: ∀a, b ∈ R, a = b =⇒ b = a.

E3 Transitivity for Equality:∀a, b , c ∈ R, a = b ∧ b = c =⇒ a = c .

E4 Addition Property for Equality:∀a, b , c ∈ R, a = b =⇒ a + c = b + c .

E5 Multiplication Property of Equality:∀a, b , c ∈ R, a = b =⇒ ac = bc .


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Number Sets

Order Axioms for R

Order Axioms for R


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Number Sets

Order Axioms for R

Axioms

1 Trichotomy Axiom: ∀a, b ∈ R, one and only of the following istrue: a > b , a = b , b > a.

2 The Transitivity Axiom: Let a, b , c ∈ R. If a > b and b > c ,then a > c .

3 The Addition Axiom: Let a, b , c ∈ R. If a > b , then

a + c > b + c .

4 The Multiplication Axiom: Let a,b ,c ∈ R. If

a >

b and

c > 0, then ac > bc .


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Number Sets

Order Axioms for R

Definitions

1 A real number a is said to be positive if a > 0 and negative

if a < 0. The number 0 is considered neither positive nornegative.

2 If a and b are real numbers, we say a < b (read ”a is less thanb”) if and only if b > a.


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Number Sets

Theorems on Real Numbers

Theorem 1

∀a ∈ R,−(−a) = a.


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Number Sets


Theorem 1

∀a ∈ R,−(−a) = a.

Theorem 2: The cancellation law for addition

∀a, b , c ∈ R, a + c = b + c =⇒ a = b .


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Number Sets


Theorem 1

∀a ∈ R,−(−a) = a.

Theorem 2: The cancellation law for addition

∀a, b , c ∈ R, a + c = b + c =⇒ a = b .

Theorem 3

∀a ∈ R, a · 0 = 0.


N b S

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Number Sets


Theorem 4

∀a, b ∈ R, (−a) · b = −(ab ).


N b S t

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Number Sets


Theorem 4

∀a, b ∈ R, (−a) · b = −(ab ).

Corollary 5

∀b ∈ R, (−1)b = −b .


Number Sets

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Number Sets


Theorem 4

∀a, b ∈ R, (−a) · b = −(ab ).

Corollary 5

∀b ∈ R, (−1)b = −b .

Corollary 6

(−1)(−1) = 1


Number Sets

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Number Sets


Theorem 7

∀a, b ∈ R, (−a)(−b ) = ab .


Number Sets

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Number Sets


Theorem 7

∀a, b ∈ R, (−a)(−b ) = ab .

Theorem 8

∀a, b ∈ R,−(a + b ) = (−a) + (−b ).


Number Sets

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Number Sets


Theorem 7

∀a, b ∈ R, (−a)(−b ) = ab .

Theorem 8

∀a, b ∈ R,−(a + b ) = (−a) + (−b ).

Theorem 9

∀a, b ∈ R,∃! solution to the equation a + x = b .


Number Sets

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Theorem 10

∀0 = a ∈ R,11a

= a.


Number Sets

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Theorem 10

∀0 = a ∈ R,11a

= a.

Theorem 11: The cancellation law for multiplication

∀a, b , 0 = c ∈ R, ac = bc =⇒ a = b .


Number Sets

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Theorem 10

∀0 = a ∈ R,11a

= a.

Theorem 11: The cancellation law for multiplication

∀a, b , 0 = c ∈ R, ac = bc =⇒ a = b .

Theorem 12

If a and b are real numbers such that a · b = 0, then a = 0 orb = 0.


Number Sets

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Theorem 13

If a and b are nonzero numbers, then

1

a · b =

1

a ·

1

b


Number Sets

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Theorem 13


1

a · b =

1

a ·

1

b

Theorem 14

For any real number b and nonzero real number a, there is aunique solution to the equation a · x = b .


Number Sets

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Theorem 13


1

a · b =

1

a ·

1

b

Theorem 14

For any real number b and nonzero real number a, there is aunique solution to the equation a · x = b .

Theorem 15

∀b ∈ R,b

1 = b .


Number Sets

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Theorem 16

∀0 = a ∈ R,a

a = 1.


Number Sets

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Theorem 16

∀0 = a ∈ R,a

a = 1.

Theorem 17

b

a

d

c =

bd

ac .


Number Sets

Th R l N b

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Theorem 16

∀0 = a ∈ R,a

a = 1.

Theorem 17

b

a

d

c =

bd

ac .

Corollary 18

0 = c ∈ R =⇒ b

a =

bc

ac .

Math 11: College AlgebraNumber Sets

Th R l N b

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Theorem 19

b a

= d a

=⇒ b = d .

b

a =

d

c =⇒ bc = ad .



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Theorem 19

b a

= d a

=⇒ b = d .

b

a =

d

c =⇒ bc = ad .

Theorem 20

b

a = 0 =⇒

1b

a

= a

b .



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Theorem 19

b a

= d a

=⇒ b = d .

b

a =

d

c =⇒ bc = ad .

Theorem 20

b

a = 0 =⇒

1b

a

= a

b .

Theorem 21

b

a = 0 =⇒

d

c

b

a

= da

cb .



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Theorem 22

−b

a = −

b

a

and−b

−a

= b

a



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Theorem 22

−b

a = −

b

a

and−b

−a

= b

a

Theorem 23

b

a +

d

c =

bc + ad

ac



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Theorem 22

−b

a = −

b

a

and−b

−a

= b

a

Theorem 23

b

a +

d

c =

bc + ad

ac

Theorem 24

The set of positive real numbers is closed under addition.



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Theorem 25The set of positive real numbers is closed under multiplication.



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Theorem 26

If a is a positive real number, then -a is negative. If a is negative,then -a is positive.



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Theorem 26

If a is a positive real number, then -a is negative. If a is negative,

then -a is positive.

Theorem 26

If a > b , then −b > −a.



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Theorem 26

If a is a positive real number, then -a is negative. If a is negative,

then -a is positive.

Theorem 26

If a > b , then −b > −a.

Theorem 27

For all a in R, either a2 = 0 or a2 > 0.



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Theorem 28

1 > 0



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Theorem 28

1 > 0

Theorem 29If a, b and c are real numbers and a > b and c < 0, then ac < bc .



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Theorem 28

1 > 0

Theorem 29If a, b and c are real numbers and a > b and c < 0, then ac < bc .

Theorem 30

If a > 0, then 1a > 0.



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Thank you for listening!♥



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Reymart S. Lagunero

Department of Mathematics and Computer ScienceUniversity of the Philippines Baguio

February 2, 2015

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Documents

Number Sets, Field Axioms and Order Axioms