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Math Help - Can anyone help me with 2 homework questions?

  1. #1
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    Can anyone help me with 2 homework questions?

    Been stuck on these two for a while now. Would anyone be able to help me with them? Thanks!

    #1.) Prove by contradiction that for all n ∈ Z, if n^3 + 5 is odd, then n is even

    #2.) Prove using a direct proof that for every rational number x, if x
    0, then 1/x is rational
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  2. #2
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    Re: Can anyone help me with 2 homework questions?

    1)
    Assume n is not even. Set n=2k+1 where k is a natural number
    Show that the assumption is false because n^3 +5 cannot be even when n=2k+1.

    2)
    Let x=\frac{a}{b}
    Show that 1/x can be written as a quotient.
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  3. #3
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    Re: Can anyone help me with 2 homework questions?

    Quote Originally Posted by Shakarri View Post
    1)
    Assume n is not even. Set n=2k+1 where k is a natural number
    Show that the assumption is false because n^3 +5 cannot be even when n=2k+1.

    2)
    Let x=\frac{a}{b}
    Show that 1/x can be written as a quotient.
    Thank you!!
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  4. #4
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    Re: Can anyone help me with 2 homework questions?

    Quote Originally Posted by mgk501 View Post
    Been stuck on these two for a while now. Would anyone be able to help me with them? Thanks!

    #1.) Prove by contradiction that for all n ∈ Z, if n^3 + 5 is odd, then n is even

    #2.) Prove using a direct proof that for every rational number x, if x
    0, then 1/x is rational
    It's easier to prove the first question using the contrapositive. Your statement is equivalent to saying if n is odd, then \displaystyle n^3 + 5 is even. If n is odd, we can write it as n = 2m + 1. So

    \displaystyle \begin{align*} n^3 + 5 &= \left( 2m + 1 \right) ^3 + 5 \\ &= \left( 2m \right) ^3 + 3 \left( 2m \right) ^2 + 3 \left( 2m \right) + 1 + 5 \\ &= 8m^3 + 12m^2 + 6m + 6 \\ &= 2 \left( 4m^3 + 6m^2 + 3m + 3 \right) \end{align*}

    which is clearly even. Q.E.D.
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    Re: Can anyone help me with 2 homework questions?

    "Proving the contrapositive" is a proper subset of "proof by contradiction".
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    Re: Can anyone help me with 2 homework questions?

    Quote Originally Posted by HallsofIvy View Post
    "Proving the contrapositive" is a proper subset of "proof by contradiction".
    I don't see how that could possibly be. In my example I did NOT reach a contradiction...
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