# Integer Axioms Flashcards

Terms Definitions
 Closure a, b are Z or N: a+b and a*b are also integers Definition of < If a, b are Z, a0 (positive) Positive Integer If a is a positive Z: then a>=1 Reflexive Property of Equality a is Z: a=a Symmetric Property of Equality a, b are Z: if a=b then b=a Transitive Property of Equality a, b, c are Z: if a=b and b=c then a=c Additive Property of Equality a, b, c are Z: if a=b then a+c=b+c Multiplicative Property of Equality a, b, c are Z: if a=b then a*c=b*c Transitive Property of Inequality a, b, c are Z: if a b|a 3) a, b, c are Z: if a|b and b|c then a|c 4) a, b, c are Z: if a|bc then a|b or a|c Negativity Properties 1) a is Z: -(-a) = a 2) -a = (-1)*a Weak Induction Let S be a subset of N satisfying: 1) 1 is S 2) If n is S then n+1 is S Strong Induction Let S be a subset of N satisfying: 1) 1 is S 2) If every natural # k < n+1 is in S, then n+1 is in S Well-Ordering Principle Let S be a nonempty subset of N, then S has a least element L Theorem 6 If n is a positive integer, then n>=1 (There is no integer n satisfying 0 0: we can find q, r are Z with a=bq+r, where 0<=r
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