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Math Help - Simple Contrapositive Proof

  1. #1
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    Simple Contrapositive Proof

    Let n \in \mathbb{Z} be an integer with n \ge 0. State and prove the contrapositive of the
    following statement: If n^2 is an odd number, then n is odd.

    I can prove this by substituting n for 2m and proving n^2 is even. but am unsure if this is the final extent to what i need to proof, how far do i need to go to prove this, should i give more examples?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Stylis10 View Post
    Let n \in \mathbb{Z} be an integer with n \ge 0. State and prove the contrapositive of the
    following statement: If n^2 is an odd number, then n is odd.

    I can prove this by substituting n for 2m and proving n^2 is even. but am unsure if this is the final extent to what i need to proof, how far do i need to go to prove this, should i give more examples?
    The contrapositive is that if n^2 is not odd (even) then n is not odd (even). Prove that (try contradiction).
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  3. #3
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    Quote Originally Posted by Drexel28 View Post
    The contrapositive is that if n^2 is not odd (even) then n is not odd (even). Prove that (try contradiction).
    But then am i not just going back and provign the original statement and therfore not actually proving it through contrapositive?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Stylis10 View Post
    But then am i not just going back and provign the original statement and therfore not actually proving it through contrapositive?
    No. It is quite different.
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  5. #5
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    ok, i think i understand, i have come up with this:

    n must be even for n^2 to be even,
    let n = 2m+1
    (2m+1)^2 = 4m^2+4m+1
    = 2Q + 1 which is odd, therefore n must be even for n^2 to be even and hence n must be odd for n^2 to be odd.

    or have i just done my own thing here and gone off on a tangent?
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  6. #6
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    Quote Originally Posted by Drexel28 View Post
    The contrapositive is that if n^2 is not odd (even) then n is not odd (even). Prove that (try contradiction).
    I dont think that is right. Read this.

    The contrapositive of "If n^2 is an odd number, then n is odd" is "If n is even then n^2 is even".

    So Stylis10, you were right in your first post when you explained the proof in the last line.
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