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Math Help - Discrete Mathematics Problem (Proofs)

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    Discrete Mathematics Problem (Proofs)

    Prove for all M and N, if M and M-N are even, then N is even.
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    Quote Originally Posted by sderosa518 View Post
    Prove for all M and N, if M and M-N are even, then N is even.
    Assume, by contradiction, that N is odd. Then, N=2k+1 for some k \in \mathbb{N}
    We also know that M = 2r for some r \in \mathbb{N}. Then, M-N = 2r - (2k+1) = 2(r-k) + 1 which is an odd number, thus M-N is odd, in contradiction, and so N is even.

    ..

    Or:

    N = M +  (-)(M - N)

    And the sum of two even integers is even.
    Last edited by Defunkt; October 9th 2009 at 01:55 PM.
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    Quote Originally Posted by sderosa518 View Post
    Prove for all M and N, if M and M-N are even, then N is even.

    Do you know that sum or substraction of even numbers is even? If you can't use (or don't know) this then prove it: it's trivial when we characterize an even number as 2k , where k is an integer.

    Well, since N = M - (M-N) we're done.



    Tonio
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    Quote Originally Posted by sderosa518 View Post
    Prove for all M and N, if M and M-N are even, then N is even.
    Or simply, if M is even then M= 2k for some integer k. If M-N is even, the M-N= 2j for some integer j. N= M-(M-N)= 2k- 2j= 2(j-k).
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