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Math Help - tautology problem

  1. #1
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    tautology problem

    Determine weather (not p^(p->q))-> not p.

    I don't know where to start. The only way I know is use truth table.
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  2. #2
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    Edit: Are you supposed to do this without truth tables?
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  3. #3
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    yes. without truth table
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  4. #4
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    Check out a Java applet at izyt.com - Boolean Logic: Applet. It not only computes the simplified expression but also lists the steps done.
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  5. #5
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    Hello, questionboy!

    I made up names for these rules:

    . . \begin{array}{cccc}(p \to q) \;=\;\sim\!p \vee q && \text{ADI} & \text{(Alternate De{f}inition of Implication)}\\ \\<br /> <br />
p \:\vee \sim\!p \;=\;T && \text{Rule [1]} \\ \\<br /> <br />
T \vee p \;=\;T && \text{Rule [2]} \end{array}



    Tautology? . \bigg[\sim\!p \wedge (p\to q)\bigg]\;\to\;\sim\! p

    \begin{array}{ccc}\bigg[\sim\!p \wedge (\sim\!p \vee q)\bigg] \;\to\;\sim\!p && \text{ADI} \\ \\ \sim\bigg[\sim\!p \wedge (\sim\!p\vee q)\bigg] \:\vee\:\sim\!p && \text{ADI} \\ \\ \bigg[p \;\vee \sim(\sim\!p \vee q)\bigg]\:\vee\:\sim\!p && \text{DeMorgan} \end{array}

    \begin{array}{cccc} p \;\vee \bigg[\sim(\sim\!p \vee q) \;\vee \sim\!p\bigg] && \quad\text{Assoc.} \\ \\ <br />
p \;\vee \bigg[\sim\!p \;\vee \sim(\sim\!p \vee q)\bigg] && \quad\text{Comm.} \\ \\ \bigg[p \;\vee \sim\!p\bigg] \;\vee \sim(\sim\!p \vee q) && \quad\text{Assoc.} \\ \\ T \;\vee \sim(\sim\!p \vee q) && \quad\text{Rule [1]} \end{array}

    . . . . . . \begin{array}{cccc} T & \qquad\qquad\qquad\quad\;\;\text{Rule [2]}\end{array}


    It is a tautology.

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  6. #6
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    Hello!

    Quote Originally Posted by questionboy View Post
    Determine weather (not p^(p->q))-> not p.

    I don't know where to start. The only way I know is use truth table.
    1) \neg p\wedge (p\rightarrow q)...................assumption to start a conditional proof
    2) \neg p..........................1, by using Conjunction Elimination
    3) [\neg p\wedge (p\rightarrow q)]\rightarrow \neg p.................1-2, by using the rule of conditional proof

    Best wishes,
    Seppel
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