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Review Axioms and Properties handout.

Commutative Property for Addition (CPA) : 7 + 4 = 4 + 7

Commutative Property for Multiplication (CPM) : 7 • 4 = 4 • 7

Associative Property for Addition (APA) : (7 + 5) + 9 = 7 + (5 + 9)

Associative Property for Multiplication (APM) : (7 • 5) • 9 = 7 • (5 • 9)

Distributive Property for Multiplication Over Addition (DPMA) :4(x + 5) = 4 • x + 4 • 5

Additive Identity : x + 0 = x

Multiplicative Identity : x • 1 = x

Additive Inverse : 7 + (-7) = 0

Multiplicative Inverse :

Property for Multiplying by -1 (PM-1) : 4 • -1 = -4, -6 • -1 = 6

Property for Multiplying by Zero (PMZ) : x • 0 = 0, 0 • x = 0

Definition of Division :

Definition of Subtraction : 7 + (-4) = 7 – 4

Reflexive Property : 14 = 14

Symmetric Property : If 7 + 5 = 12, then 12 = 7 + 5

Transitive Property : If a = b and b = c, then a = c

Addition Property of Equality (APE) : If x = y, then x + z = y + z. You can add the same number to

both sides of an equation and not affect the solution.

Multiplication Property of Equality (MPE) : If x = y, then xz = yz, z ≠ 0. You can multiply both sides

of an equation by the same non-zero number.

Converse : A statement is true “both ways” you read it. For example : If a figure is a triangle, then

the sum of the angles is 180º. The converse reads : If the sum of the angles is 180º, then the figure

is a triangle.

Substitution : If a = b + c, then (usually later in the proof), d = a, then d = b + c.

Trichotomy (Comparison Property) : Given any two real numbers a and b, exactly one of the

following is true : a > b, a < b, or a = b.

1) Prove : If x + z = y + z, then x = y.

STATEMENTREASON

a) x + z = y + z

a) Given

b) x + z + (-z) = y + z + (-z)

b) APE

c) x + [z + (-z)] = y + [z + (-z)]

c) APA

d) x + 0 = y + 0

d) Inverse

e) x = y

e) Identity

We just proved the converse of APE.

2) Prove : If x + b = a, then x = a + (-b)

STATEMENTREASON

a) x + b = a

a) Given

b) APE

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

c) x + 0 = x + (-b)

c) Inverse

d) x = a + (-b)

d) Identity

Usually, identity follows inverse.

3) Prove : If ab = b and b ≠ 0, then a = 1.

STATEMENTREASON

a) ab =b and b ≠ 0

a) Given

b)

b) MPE

c)

c) APM

d) a • 1 = 1

d) Inverse

e) a = 1

e) Identity

Prove : If ax + b = c and a ≠ 0, then

STATEMENTREASON

a) ax + b = c, a ≠ 0

a) Given

b) ax + b + (-b) = c + (-b)

b) APE

c) ax + [b + (-b)] = c + (-b)

c) APA

d) ax + 0 = c + (-b)

d) Inverse

e) ax = c + (-b)

e) Identity

f)

f) MPE

g)

g) APM

h) Inverse

h)

i)

i) Identity

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