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, a<b if a+n=b for some N, n.
Trichotomy
If a is Z, then one of the following is true: a<0 (negative), a=0, a>0 (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 and b<c then a<c.
Additive Property of Inequality
a, b, c are N: if a<b then a+c<b+c (not true in Z)
Multiplicative Property of Inequality
a, b, c are N: if a<b then a*c<b*c
Substitution Principle
You can substitute equals for equals in any expression without changing its truth or falsity.
Associative Property of Addition
a, b, c are Z: (a+b)+c=a+(b+c)
Associative Property of Multiplication
a, b, c are Z: (a*b)*c=a*(b*c)
Commutative Property of Addition
a, b are Z: a+b=b+a
Commutative Property of Multiplication
a, b are Z: a*b=b*a
Distributive Property
a, b, c are Z: a*(b+c)=a*b+a*c
Additive Identity
a is Z: a+0=0+a=a
Multiplicative Identity
a is Z: a*1=1*a=a
Additive Inverse
a is Z: there exists an integer -a satisfying a+(-a)=(-a)+a=0
Maryam's Theorem
a is Z: a*0=0
Cancellation Under Addition
a, b, c are Z: if a+c=b+c then a=b
Cancellation Under Multiplication
a, b, c are Z: if a*c=b*c then a=b
Divisibility Properties
1) a is Z: a|a
2) a, b are Z: if a=b, b=/0 then a|b --> 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<n<1)
Division Algorithm
a, b are Z, b > 0: we can find q, r are Z with a=bq+r, where 0<=r<b
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