# Ch.1 Properties

Definition of Subtraction a – b = a + (-b)
Definition of Division a ? b = a/b = a • 1/b, b?0
Distributive Property for Subtraction a(b – c) = ab – ac
Multiplication by 0 0 • a = -a
Multiplication by -1 -1 • a = -a
Opposite of a Sum -(a + b) = -a + (-b)
Opposite of a Difference -(a -b) = b-a
Opposite of a Product -(ab) = -a • b = a •(-b)
Opposite of an Opposite -(-a) = a
Reflexive Property 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, then a + c = b + c
Subtraction Property of Equality If a = b, then a – c = b – c
Multiplication Property of Equality If a = b, then ac = bc
Division Property of Equality If a = b and c ? 0, then a/c = b/c
Substitution Property of Equality If a = b, then b may be substituted for a in any expression to obtain an equivalent expression
Transitive Property of Inequality If a ? b and b ? c, then a ? c
Addition Property of Inequality If a ? b, then a + c ? b + c
Subtraction Property of Inequality If a ? b, then a – c ? b – c
Multiplication Property of Inequality If a ? b and c > 0, then ac ? bc. If a ? b and c < 0, then ac ? bc.
Division Property of Inequality If a ? b and c > 0, then a/c ? b/c. If a ? b and c < 0, then a/c ? b/c.
Closure of Addition a + b is a real number
Closure of Multiplication ab is a real number
Communtative of Addition a + b = b + a
Communtative of Multiplication ab = ba
Associative of Addition (a + b) + c = a + (b +c)
Associative of Multiplication (ab)c = a(bc)`
Identity of Addition a + 0 = a, 0 + a = a
Identity of Multiplication a • 1 = a, 1• a = a
Inverse of Addition a + (-a) = 0
Inverse of Multiplication a • 1/a = 1, a ? 0
Distributive a(b + c) = ab + ac