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Thread: 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 $\displaystyle n\in\mathbb{Z}$ and let $\displaystyle 3n+2$ be even.

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

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

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

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


    Can you give the second part b try
    Last edited by Plato; Nov 8th 2009 at 02:20 PM.
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