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Math Help - Proofs can't figure it out

  1. #1
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    Proofs can't figure it out

    I am completely lost on how to do this.
    Prove that an integer n is even if and only if n+1 is odd

    thanks in advance for any help
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  2. #2
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    For starters make n+1 odd as follows

    2(n+1)+1 now show the previous number is even.
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  3. #3
    Member oldguynewstudent's Avatar
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    even odd proof

    T.P. n is even iff n+1 is odd

    Assume n is even then let n = 2k for some k element of Z (definition of even)

    Then n+1 = 2k + 1, but 2k + 1 is the definition of an odd number. This proves the forward direction now reverse it.

    Let n + 1 be odd then by definition of an odd number n + 1 = 2k + 1 for an arbitrary k element of Z

    Now subtract 1 from both sides and you have n = 2k which by definition is even.

    This completes the proof in both directions.
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