number sets, field axioms and order axioms
TRANSCRIPT
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Math 11: College Algebra
Math 11: College Algebra
Reymart S. Lagunero
Department of Mathematics and Computer ScienceUniversity of the Philippines Baguio
February 2, 2015
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Math 11: College Algebra
Number Sets
Number Sets
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
The Real Number System
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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}.
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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.
Math 11: College Algebra
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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.
Math 11: College Algebra
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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}.
Math 11: College Algebra
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Math 11: College Algebra
Number Sets
The Algebra of Counting Numbers
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
.
Math 11: College Algebra
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Math 11: College Algebra
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 .
Math 11: College Algebra
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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.
Math 11: College Algebra
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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 .
Math 11: College Algebra
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Number Sets
Order Axioms for R
Order Axioms for R
Math 11: College Algebra
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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 .
Math 11: College Algebra
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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.
Math 11: College Algebra
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Number Sets
Theorems on Real Numbers
Theorem 1
∀a ∈ R,−(−a) = a.
Math 11: College Algebra
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Number Sets
Theorems on Real Numbers
Theorem 1
∀a ∈ R,−(−a) = a.
Theorem 2: The cancellation law for addition
∀a, b , c ∈ R, a + c = b + c =⇒ a = b .
Math 11: College Algebra
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Number Sets
Theorems on Real Numbers
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.
Math 11: College Algebra
N b S
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Number Sets
Theorems on Real Numbers
Theorem 4
∀a, b ∈ R, (−a) · b = −(ab ).
Math 11: College Algebra
N b S t
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Number Sets
Theorems on Real Numbers
Theorem 4
∀a, b ∈ R, (−a) · b = −(ab ).
Corollary 5
∀b ∈ R, (−1)b = −b .
Math 11: College Algebra
Number Sets
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Number Sets
Theorems on Real Numbers
Theorem 4
∀a, b ∈ R, (−a) · b = −(ab ).
Corollary 5
∀b ∈ R, (−1)b = −b .
Corollary 6
(−1)(−1) = 1
Math 11: College Algebra
Number Sets
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Number Sets
Theorems on Real Numbers
Theorem 7
∀a, b ∈ R, (−a)(−b ) = ab .
Math 11: College Algebra
Number Sets
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Number Sets
Theorems on Real Numbers
Theorem 7
∀a, b ∈ R, (−a)(−b ) = ab .
Theorem 8
∀a, b ∈ R,−(a + b ) = (−a) + (−b ).
Math 11: College Algebra
Number Sets
8/9/2019 Number Sets, Field Axioms and Order Axioms
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Number Sets
Theorems on Real Numbers
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 .
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 10
∀0 = a ∈ R,11a
= a.
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 10
∀0 = a ∈ R,11a
= a.
Theorem 11: The cancellation law for multiplication
∀a, b , 0 = c ∈ R, ac = bc =⇒ a = b .
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
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.
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 13
If a and b are nonzero numbers, then
1
a · b =
1
a ·
1
b
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 13
If a and b are nonzero numbers, then
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 .
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 13
If a and b are nonzero numbers, then
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 .
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 16
∀0 = a ∈ R,a
a = 1.
Math 11: College Algebra
Number Sets
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Theorems on Real Numbers
Theorem 16
∀0 = a ∈ R,a
a = 1.
Theorem 17
b
a
d
c =
bd
ac .
Math 11: College Algebra
Number Sets
Th R l N b
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Theorems on Real Numbers
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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Theorems on Real Numbers
Theorem 19
b a
= d a
=⇒ b = d .
b
a =
d
c =⇒ bc = ad .
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorems on Real Numbers
Theorem 19
b a
= d a
=⇒ b = d .
b
a =
d
c =⇒ bc = ad .
Theorem 20
b
a = 0 =⇒
1b
a
= a
b .
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorems on Real Numbers
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 .
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorems on Real Numbers
Theorem 22
−b
a = −
b
a
and−b
−a
= b
a
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorems on Real Numbers
Theorem 22
−b
a = −
b
a
and−b
−a
= b
a
Theorem 23
b
a +
d
c =
bc + ad
ac
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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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.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorem 25The set of positive real numbers is closed under multiplication.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorem 25The set of positive real numbers is closed under multiplication.
Theorem 26
If a is a positive real number, then -a is negative. If a is negative,then -a is positive.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorem 25The set of positive real numbers is closed under multiplication.
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.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorem 25The set of positive real numbers is closed under multiplication.
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.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Theorem 28
1 > 0
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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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 .
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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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.
Math 11: College AlgebraNumber Sets
Theorems on Real Numbers
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Thank you for listening!♥
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Theorems on Real Numbers
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Math 11: College Algebra
Reymart S. Lagunero
Department of Mathematics and Computer ScienceUniversity of the Philippines Baguio
February 2, 2015