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Math Help - Proof by Contraposition... (and background hypothesis PLEASE HELP!)

  1. #1
    Junior Member mremwo's Avatar
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    Proof by Contraposition... (and background hypothesis PLEASE HELP!)

    Prove by using proof by contraposition (and background hypothesis) that "the product of a non-zero rational number m and irrational number n is irrational....

    im supposed to use contraposition and background hypothesis to change this statement to...

    in the background ---> (m not equal to 0): and the proof ---> (if mn is rational, then n is rational)

    BUT i do not know where/if im supposed to incorporate the "background" portion of my hypothesis into my proof (because i do know that you cant prove "if mn is rational then n is rational" since such a proof does not exist.)
    Last edited by mremwo; October 4th 2010 at 01:59 PM. Reason: EDIT... i meant to put (m not equal to zero and rational)
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  2. #2
    MHF Contributor
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    You said yourself that without the hypothesis that m <> 0 you can't prove the claim, so you have to use it. Is there any reason not to?

    Here is an outline: you are proving a statement of the form A -> (B -> (C -> D)) where A is "m is rational", B is "m <> 0", C is "n is irrational" and D is "mn is irrational". This is equivalent to A -> (B -> (~D -> ~C)). In proving the latter, you assume A, B and ~D and prove ~C.

    To explain the formula A -> (B -> (C -> D)): this is equivalent to A /\ B /\ C -> D. Indeed, saying "if A, then: if B, then C" is the same as saying "if A and B, then C". Formally this can be seen using truth tables.
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