Axioms

Closure Axiom of Addition

a+b=c, where c is a unique real number

Closure Axiom of Multiplication

ab=c, where c is a unique real number

Commutative Axiom of Addition

a+b=b+a

Commutative Axiom of Multiplication

ab=ba

Associative Axiom of Addition

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

Associative Axiom of Multiplication

a(bc)=(ab)c

Identity Axiom of Addition

a+0=0+a=a, where 0 is unique

Identity Axiom of Multiplication

a(1)=(1)a=a, where 1 is unique

Inverse Axiom of Addition

a+(-a)=(-a)+a=0 where -a is unique

Inverse Axiom of Multiplication

a(1/a)=(1/a)a=1 where 1/a is unique, a doesn't = 0

Distributive Axiom

a(b+c)=ab+ac

Reflexive Axiom of Equality

a=a

Symmetric Property of Equality

If a=b, then b=a

Transitive Property of Equality

If a=b and b=c, then a=c

Addition Property of Equality

If a=b and c=d then a+c=b+d

Multiplication Property of Equlaity

If a=b and c=d then ac=bd

Definition of Subtraction

a-b=a+(-b)

Definition of Division

a/b=a(1/b), b doesn't = 0

Definition of Greater Than

a>b iff a-b>0

Definition of Less Than

a

Addition Axiom of Inequality (>)

If a>b then a+c>b+c

Multiplication Axiom of Inequality (>)

If a>b and c>0 then ac>bc

If a>b and c<0 then ac

Transitive Axiom of Inequality (>)

If a>b and b>c then a>c

Multiplication Axiom of Inequality (<)

If a0 then ac0 then ac>bc