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Proof by Contradiction

  1. Mar 15, 2016 #1
    1. The problem statement, all variables and given/known data

    Prove by contradiction that if b is an integer such that b does not divide k for every natural number k, then b=0.
    2. Relevant equations


    3. The attempt at a solution
    I know that proof by contradiction begins by assuming the false statement: If b is an integer such that b does not divide k for every kεℕ, then b≠0, which is equivalent to "there exists an integer b such that b does not divide k and b≠0, for every kεℕ. But I'm not sure how to proceed from here.
     
  2. jcsd
  3. Mar 15, 2016 #2

    mfb

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    Staff: Mentor

    That is not the opposite statement.
    "If a pen is green, then it is my pen" is wrong, but "if a pen is green, then it is not my pen" is also wrong (because I own some green pens, but not all).
     
  4. Mar 16, 2016 #3

    HallsofIvy

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    Also "b does not divide k for every natural number k" is itself ambiguous. It could be read as "b does not divide any natural number" but here, I think, is intended to say "there exist a natural number, k, that b does not divide". The "contradiction" would just be "there exist b, not equal to 0, such that b does not divide any natural number, k." Given that, what can you say about k= 2b?
     
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