L05_proofs_by_contradiction

L05_proofs_by_contradiction - 1-1Proof by Smallest...

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Unformatted text preview: 1-1Proof by Smallest CounterexampleDefinitions:•log2(n)isxsuch that2x=n.blog2(n)cis the uniqueis.t.2i≤n <2i+1e.g.blog2(2)c= 1,blog2(3)c= 1,blog2(4)c= 2blog2(31)c= 4,blog2(32)c= 5,blog2(33)c= 51-2Proof by Smallest CounterexampleDefinitions:•log2(n)isxsuch that2x=n.blog2(n)cis the uniqueis.t.2i≤n <2i+1•Prime factorization ofnis the representation ofnasmultiplication of a list of primes.e.g.12 = 2×2×3,6! = 2×2×2×2×3×3×5e.g.blog2(2)c= 1,blog2(3)c= 1,blog2(4)c= 2blog2(31)c= 4,blog2(32)c= 5,blog2(33)c= 51-3Proof by Smallest CounterexampleDefinitions:•log2(n)isxsuch that2x=n.blog2(n)cis the uniqueis.t.2i≤n <2i+1•Prime factorization ofnis the representation ofnasmultiplication of a list of primes.e.g.12 = 2×2×3,6! = 2×2×2×2×3×3×5•DefineSIZE(n)to be thenumber of prime factorsinprime factorization ofn.e.g.SIZE(12) = 3,SIZE(6!) =SIZE(720) = 7e.g.blog2(2)c= 1,blog2(3)c= 1,blog2(4)c= 2blog2(31)c= 4,blog2(32)c= 5,blog2(33)c= 52-1Proof by Smallest CounterexampleTheorem:For any positive integern,SIZE(n)≤ blog2(n)c.2-2Proof by Smallest CounterexampleTheorem:For any positive integern,SIZE(n)≤ blog2(n)c....
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This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

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L05_proofs_by_contradiction - 1-1Proof by Smallest...

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