# PracticeProofsTest2.pdf - Discrete Math Proof Problems...

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Discrete MathProof ProblemsInductions1)For all integers3n,(2)(3)3452nnn.2)A sequence123,,,...aaais defined recursively as follows:1041 for all integers12kkaakaProve that7 413nnafor all integers0n.3)A function is defined recursively as follows:41( )823 (1)(2)2nS nnS nS nnProve that( )S nis divisible by 4for all integers1n.4)Let f:NNbe a function so that f(a + b) = f(a) * f(b) and f(1) = 2.Prove f(n) = 2nfor allinteger n inN.5)Define a set X recursively asB.2XR.If, then4xXxXProve that every element of X is even.6)Define a recursive function PRD on the set of lists of real numbers by;1212.( ),if.( )()()if,PRD LxLxPRD LPRD LPRD LLL LBRThe set of lists on the real numbers is defined by1212.for all.,for all lists,LxxLL LL LBRRSet Proofs:7)For all sets A, B,ifAB, thenABA8)For all sets A, B and C,()()ABCACBC

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