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Math Help - How to prove this

  1. #1
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    How to prove this

    How to prove ((p→q)∧p) → q is a tautology using equivalence laws?
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  2. #2
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    Re: How to prove this

    As per this thread, could you please list what equivalences you're allowed to use? What are your inference rules?
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  3. #3
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    Re: How to prove this

    Hello, MathsNewbie0811!

    Prove . [(p\to q)\wedge p]\:\to\; q is a tautology using equivalence laws.

    \begin{array}{ccccccccc} 1. & [( p\to q) \wedge p] \:\to\:q && 1. & \text{Given} \\2. & [(\sim\!p \vee q) \wedge p] \:\to\:q && 2. & \text{Def. Impl'n} \\ 3. & [(\sim\!p \wedge p) \vee (q \wedge p)] \:\to\;q && 3.&  \text{Distr.} \\ 4. & [F \vee (q\vee p)] \:\to\:q && 4. & s \:\wedge \sim\!s \:=\:F \\ 5. & \sim(q \wedge p) \vee q && 5. &\text{Def. Impl'n} \\ 6. & (\sim\!q \:\vee \sim\!p) \vee q && 6. & \text{DeMorgan} \\ 7. & (\sim\!q \vee q) \vee p && 7. & \text{Comm, Assoc.} \\ 8. & T \:\vee \sim\!p && 8. & s \:\vee \sim\!s \:=\:T \\ 9. & T && 9. & T \vee s \:=\:T  \end{array}

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  4. #4
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    Re: How to prove this

    Should line 8 be T \vee p

    Just making sure I understand. From line 7 you apply the fact that q\vee \lnot q \equiv T

    Why did the p change?
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  5. #5
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    Re: How to prove this

    Thank you.
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  6. #6
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    Re: How to prove this

    Quote Originally Posted by terrorsquid View Post
    Should line 8 be T \vee p
    Line 8 is correct, but line 7 should be (~q \/ q) \/ ~p.
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