Axioms and Properties for Multiplication and Addition

if x = y AND y = z, then x = z

Transitive Axiom of Equality

x * 1 = x

Multiplicative Identity Axiom

x * 1/x = 1

Multiplicative Inverse Axiom

-1 * x = -x

Multiplicative Property of -1

x * 0 = 0

Multiplicative Property of 0

x = x

Reflexive Axiom of Equality

if x = y, then y = x

Symmetric Axiom of Equality

if x = y, then x + z = y + z

Addition Property of Equality

x + 0 = x

Additive Identity Axiom

x + -x = 0

Additive Inverse Axiom

(x + y) + z = x + (y + z)

Associative Axiom for Addition

(xy) z = x (yz)

Associative Axiom for Multiplication

x + y = y + x

Commutative Axiom for Addition

xy = yx

Commutative Axiom for Multiplication

x(y +z)= xy + xz

Distributive Axiom for Multiplication over Addition

if x = y, then xz = yz

Multiplication Property of Equality