if a>b and b>c, then a>c

Transitive Property of Inequality

if a=b, then a**c=b**c

Multiplication Property of Equality

-(-a) = a

The -(-a) = a property

1/(1/a) = a

The 1/(1/a) = a property

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

Property of the Opposite of a Sum

a/b = a*(1/b)

Definition of Division

a-b=a+(-b)

Definition of Subtraction

The abs. value of a = a if a >= 0

The abs. value of a = -a if a < 0

The abs. value of a = -a if a < 0

Definition of Absolute Value

A Set S is closed under operation # if for every a,b that are members of set S, a#b is also a member of S.

Definition of Closure

a=a

Reflexive Property of Equality

if a=b, then b=a

Symmetric Property of Equality

if a=b and b=c, then a=c

Transitive Property of Equality

if a=b, then a can be substituted for b at any time

Substitution

if a=b, then a+c=b+c

Addition Property of Equality

if a=b, then a-c=b-c

Subtraction Property of Equality

if a=b, then ac = bc

Multiplication Property of Equality

if a=b, then a/c=b/c

Division Property of Equality

a+b=b+a

Commutative Property of Addition

ab=ba

Commutative Property of Mult.

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

Associative Property of Addition

(ab)c=a(bc)

Associative Property of Mult.

x+0=x

Identity Property of Addition

x*1=x

Identity Property of Multiplication

a+(-a)=0

Additive Inverse Property

a * (1/a)=1

Multiplicative Inverse Property

a*0=0

Multiplicative Property of Zero

if ab=0, then a=0 or b=0 or both a&b=0

Zero Product Property

a(b+c) = ab + ac

Distributive Property

a-b = a+ (-b)

Definition of Subtraction

between any two rational numbers, there are many other rational numbers (true for irrationals/reals too)

Density Property

Rules for A

a > 0 --> 1/a > 0

a < 0 --> 1/a < 0

a > b --> 1/a < 1/b

* true if a and b are the same sign

a < 0 --> 1/a < 0

a > b --> 1/a < 1/b

* true if a and b are the same sign

Property of Squares

a^2 ≥ 0