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What would be the contrapositive of this statement?

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data
    The original statement is Prove that if xy and x+y are even then both x and y are even.


    2. Relevant equations



    3. The attempt at a solution
    I think it goes like "If x or y is odd then xy and x+y are odd"? I'm not too sure though because the first "and" in the hypothesis is confusing. I'm not sure if its supposed to go like"If x or y is odd then xy or x+y is odd?
     
  2. jcsd
  3. Sep 20, 2012 #2

    jbunniii

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    You want to negate the statement "xy and x+y are even". If this statement is not true, it means that either xy is not even or x+y is not even, or both. And "not even" is the same as "odd". So indeed "If x or y is odd then xy or x+y is odd" is correct.
     
  4. Sep 20, 2012 #3

    HallsofIvy

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    The statement is "If ((x is even) and (y is even)) then ((xy is even) and (x+y is even))".

    As you suggest, the contrapositive of "if A then B" is "if not B then not A". So the contrapositive of this statement is
    "if NOT ((xy is even) and (x+y is even)) then NOT ((x is even) and (y is even))".

    The thing you are missing is that "NOT A and B" is "(Not A) or (not B)".
    So the contra positive is
    "if xy is odd or x+y is odd, then either x is odd or y is odd".
     
  5. Sep 20, 2012 #4
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  6. Sep 21, 2012 #5
    I would think the sentence would have to be " If x is odd or y is odd then xy is odd or x+y is odd." I think you have it flipped around.
     
  7. Sep 21, 2012 #6

    SammyS

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    To whom are you responding?

    (We have a "QUOTE" feature to take the guess-work out of this.)

    By the Way: " If x is odd or y is odd then xy is odd or x+y is odd." is a valid contrapositive to the original statement.
     
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