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Math Help - Contrapositive Method

  1. #1
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    Question Contrapositive Method

    Plz help solving these:

    (a) Assume that  m and  n are positive integers and that  m \le n . Prove that if  m^2 = n^2 , then  m=n using contrapositive method.

    (b) Using direct proof show that the negative of any even integer is even.


    Thnx in advance..
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by cu4mail View Post
    Plz help solving these:

    (a) Assume that  m and  n are positive integers and that  m \le n . Prove that if  m^2 = n^2 , then  m=n using contrapositive method.
    to prove using the contrapositive method means the following.

    by the contrapositive method, we prove a statement P \implies Q by showing \neg Q \implies \neg P

    so start by assuming m \ne n, where does this get you? how does this cause m^2 to relate to n^2? will they be equal?

    (b) Using direct proof show that the negative of any even integer is even.
    Hint: we say an integer is even if it can be expressed as 2n for n \in \mathbb{Z}. now note that -2n = 2(-n)
    Last edited by Jhevon; January 8th 2008 at 09:18 AM. Reason: corrected an error, thanks Isomorphism!
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    to prove using the contrapositive method means the following.
    by the contrapositive method, we prove a statement P \implies Q by showing \neg P \implies \neg Q
    Shouldn't that read "by the contrapositive method, we prove a statement P \implies Q by showing \neg Q \implies \neg P"
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Isomorphism View Post
    Shouldn't that read "by the contrapositive method, we prove a statement P \implies Q by showing \neg Q \implies \neg P"
    yes, indeedy! thanks. i'll fix it (hehe, i must be dyslexic or something, i had to read what you wrote like three times before i saw the difference between what i wrote and what you wrote)
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    to prove using the contrapositive method means the following.

    by the contrapositive method, we prove a statement P \implies Q by showing \neg Q \implies \neg P

    so start by assuming m \ne n, where does this get you? how does this cause m^2 to relate to n^2? will they be equal?

    Hint: we say an integer is even if it can be expressed as 2n for n \in \mathbb{Z}. now note that -2n = 2(-n)
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  6. #6
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    and are positive integers and that . Prove that if , then using contrapositive method.
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  7. #7
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    and are positive integers and that . Prove that if , then using contrapositive method.
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by adnan View Post
    and are positive integers and that . Prove that if , then using contrapositive method.
    Quote Originally Posted by adnan View Post
    and are positive integers and that . Prove that if , then using contrapositive method.
    why do you keep repeating the first question? did you see the hint i gave?
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