| A | B |
|---|
| commutative axiom | ab=ba |
| associative axiom | (ab)c=a(bc) |
| reflective property of equality | a=a |
| symmetric property of equality | a=b then b=a |
| transitive property of equality | a=b & b=c then a=c |
| identity axiom for addition | a+0=0 |
| axiom of opposites | a+(-a)=0 |
| property of the opposite of a sum | -(a+b)=(-a)+(-b) |
| definition of subtraction | a-b=a+(-b) |
| distributive axiom of multiplication w/ respect to addition | a(b+c)=ab+ac |
| distributive axiom of multiplication w/ respect to subtraction | a(b-c)=ab-ac |
| identity axiom for multiplication | a x 1=a |
| multiplicative property for 0 | a x 0=0 |
| multiplicative property for -1 | a x (-1)=(-a) |
| property of opposites in products | (-a)(b)=(-ab) or (-b)(a)=(-ab) or (-a)(-b)=ab |
| axiom of reciprocals | a x (1/a)=1 |
| property of the reciprocal of a product | 1/ab=(1/a)(1/b) |
| definition of division | a/b=a(1/b) |
| addition property of equality | a+c=b+c when a=b |
| subtraction property of equality | a-c=b-c when a=b |
| multiplication property of equality | a=b then c x a=c x b |
| division property of equality | a=b then a/c=b/c |
| inverse operations | (a+b)-b=a or (a-b)+b=a |
