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Proof by Contradiction and Contraposition

  1. Jul 29, 2009 #1
    1. The problem statement, all variables and given/known data

    For any integer n, let A(n) be the statement:
    "If n=4q+1 or n=4q+3 for some [tex]q \in Z[/tex], then n is odd."

    (a) Write down the contrapositive of A(n).
    (b) Is the contrapositive of A(n) true for all [tex]n \in N[/tex]? Give brief reasons.
    (c) Use proof by contradiction to show that the converse of A(n) is true for all [tex]n \in Z[/tex].

    2. Relevant equations



    3. The attempt at a solution

    (a)

    The contrapositive of [tex]A \Rightarrow B[/tex] is [tex]\neg B \Rightarrow \neg A[/tex], so in this case:

    A:= If n=4q+1 or n=4q+3 for some [tex]q \in Z[/tex]
    B:= n is odd

    The contrapositive would be:

    If n is even then [tex]n \neq 4q+1[/tex] and [tex]n \neq 4q+3[/tex] for all [tex]q \in Z[/tex].

    Is this right?

    (b)

    I have no idea how to answer this one.

    (c)

    The converse of the A(n) is: "If n is odd, then n=4q+1 or n=4q+3 for some [tex]q \in Z[/tex]"

    A:= n is odd
    B:= n=4q+1 or n=4q+3

    Now this is what I have to show for a proof by contradiction:
    [tex](A \Rightarrow B) \Leftrightarrow \neg (A \wedge \neg B)[/tex]

    [tex]A \wedge \neg B[/tex] = n is odd and [tex]n \neq 4q+1[/tex] or [tex]n \neq 4q+3[/tex]

    Of course, n is odd [tex]\Leftrightarrow (\exists m \in Z)(2m+1)[/tex]

    So, I'm stuck here & I'm not sure exactly how to prove that [tex]A \wedge \neg B[/tex] is false. Any help is appreciated.
     
  2. jcsd
  3. Jul 29, 2009 #2
    a) Looks good
    b) Search for a counterexample. If you can't, try proving that it's true based on the intuition gained from the search for the counterexample.
    c) Any integer m is either even or odd.
     
  4. Jul 29, 2009 #3
    About part (C), I need to prove this wrong:

    [tex]A \wedge \neg B[/tex] = "n is odd and [tex]n \neq 4q+1[/tex] or [tex]n \neq 4q+3[/tex]"

    A:= n is odd
    B:= n=4q+1 or n=4q+3

    Suppose A and [tex]\neg B[/tex] are true; n is odd. n is odd if n=2m+1 and even if n=2m. However n=4q+1 and n=4q+3 are both odd since:

    n=4q+1 = 2(2q)+1

    And 4q+3= 2(2q+1)+1

    [tex]2q+1 \in Z[/tex] and [tex]2q \in Z[/tex]. Hence [tex]\neg B[/tex] doesn't imply A, so the converse of A(n) is true for all [tex]n \in Z[/tex].

    Is this right??
     
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