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Chapter_03-4up

# Chapter_03-4up - Definitions N!Z!Q!R Number Theory and...

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Comp Sci 360 - Fall 2009 Number Theory and Proof Techniques John Hannan - Comp Sci 360 - Fall 2009 Definitions N ! Z ! Q ! R Even Odd Prime Composite John Hannan - Comp Sci 360 - Fall 2009 Proving Existentials " x # S. Q(x) Find some x in S Show existence is guaranteed By contradiction John Hannan - Comp Sci 360 - Fall 2009 Disproving Universals \$ x # S.Q(x) \$ x # S.P(x) ! Q(x) Find Counterexample

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John Hannan - Comp Sci 360 - Fall 2009 Proving Universals \$ x # S.Q(x) \$ x # S.P(x) ! Q(x) Exhaustion Contradiction Direct (Generic) Induction (later) Contrapositive John Hannan - Comp Sci 360 - Fall 2009 Examples All even numbers between 4 and 30 can be written as the sum of two primes n*(n+1) is even for all n # Z Sum of any two odd numbers is even For all n, if n is odd, n 2 is odd if n is even then (-1) n = 1 if k>0 then k 2 + 2k + 1 is composite John Hannan - Comp Sci 360 - Fall 2009 Rationals Definition Sum, Difference, Product, Quotient Reduced Form John Hannan - Comp Sci 360 - Fall 2009 Divisibility n is divisible by d iff n=dk for some integer k d|n
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Chapter_03-4up - Definitions N!Z!Q!R Number Theory and...

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