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Math Help - Show this is a tautology using a calculation

  1. #1
    Junior Member
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    Show this is a tautology using a calculation

    I've to show via a calculation that ((P => (Q v R)) ^ (Q => R)) => (P => R) is a tautology. Now, to show this, I've to show ((P => (Q v R)) ^ (Q => R)) |= (P =>R).

    I've tried the following:

    (P => (Q v R)) ^ (Q => R)
    |= { by ^ v weakening }
    P => (Q v R)
    = { by distribution }
    (P => Q) v (P => R)

    But now I'm stuck, since ((P => Q) v (P => R)) is actually a weaker proposition than (P => R).

    Obviously I'm doing something wrong. Should I apply monotonicity? Any help would be greatly appreciated!
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  2. #2
    MHF Contributor
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    Re: Show this is a tautology using a calculation

    Once you have (P => Q) v (P => R), you reason by cases. If P => Q, then together with Q => R this gives P => R. Otherwise, you have P => R directly.
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