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Math Help - Implication theorem

  1. #1
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    Implication theorem

    The following theorem has the form of an implication.

    If n is an integer and 3n+2 is even, then n is even.

    a) Prove this theorem by contradiction.
    b) Give a direct proof of this theorem.
    (Hint: it may be useful to get an equation for n in which
    the term to which n is equal also includes n itself.)
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  2. #2
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    Quote Originally Posted by kashifzaidi View Post
    The following theorem has the form of an implication.

    If n is an integer and 3n+2 is even, then n is even.

    a) Prove this theorem by contradiction.
    b) Give a direct proof of this theorem.
    (Hint: it may be useful to get an equation for n in which
    the term to which n is equal also includes n itself.)

    Let n\in\mathbb{Z} and let 3n+2 be even.

    Assume n is not even. Therefore n is odd. Therefore n=2x+1 for some x\in\mathbb{Z}

    So 3n+2=3(2x+1)+2=6x+3+2=6x+5=2(3x+2)+1

    But x is an integer, so 3x is an integer, so 3x+2 is an integer. Thefore 2(3x+2)+1=2z+1 for some z\in\mathbb{Z}

    This is the definition of an odd number, so 3n+2 is odd. But we know 3n+2 is even. This is a contradiction and therefore the assumption that n is not even is wrong. Therefore n is even


    Can you give the second part b try
    Last edited by Plato; November 8th 2009 at 03:20 PM.
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