| A | B |
|---|
| Closure axiom for addition | a + b = real number |
| Closure Axiom for multiplication | a x b = real number |
| Commutative Axiom | a + b = b + a |
| Associative Axiom | (a + b) + c = a+ (b + c) |
| Reflective Axiom of Equality | a = a |
| Symmetric Axiom of Equality | a = b, b = a |
| Transitive Axiom of Equality | If a = b, and b = c, then a = c |
| Distrubituve Axiom of Multiplication with respect to addtion | a (b + c) = ab + ac |
| Cancellation property of opposites | -(-a) = a |
| Identity axiom for addition | a + 0 = a |
| Property of the opposite of a sum | -(a + b) = -a + -b |
| Definition of subtraction | a - b = a + -b |
| Identity axiom for Multiplication | a x 1 = a |
| Multiplicative Property of zero | a x 0 = 0 |
| Mulitiplicative Property of -1 | a x -1 = -a |
| Property of opposites in products | -a x b = -ab |
| Axiom of Multiplicative Inverse | a x 1/a =1 |
| Property of the opposite of the reciprocal of a number | 1/-a = -1/a |
| Property of the reciprocal of a product | 1/ab = 1/a x 1/b |
| Definition of Divistion | a/b = a x 1/b |
