Algebra II Axioms and Properties

Associative for addition

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

Associative for multiplication

x(yz)=(xy)z

Commutative for addition

x+y=y+x

Commutative for multiplication

xy=yx

Distributive for multiplication over addition

x(y+z)=xy+xz

Additive identity

x+0=x

Multiplicative identity

x*1=x

Additive inverse

x+(-x)=0

Multiplicative inverse

x*(1/x)=1

Reflexive

x=x

Symmetric

If x=y then y=x

Transitive

if x=y and y=z, then x=z

Additive property of equality

If x=y then x+z=y+z

Additive property of inequality

if x<y then x+z<y+z etc.

Multiplicative property of equality

If x=y then xz=yz

Multiplicative property of zero

x*0=0

Multiplicative property of -1

x*-1=-x

Square root property of equality

If x=y then \sqrt{x}=\sqrt{y}

Definition of subtraction

x-y=x+(-y)

Definition of division

x/y=x*(1/y)

Substitution

x*y=xy etc.

Factoring

xz+yz=z(x+y)