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Math Help - Logical equivalences

  1. #1
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    Logical equivalences

    Use the logical equivalences to show that:

    (a) (p → (q → r)) ≡ (q → (p V r))

    (b) (p → q) & (p V q) is a contradiction (i.e. always false).

    (c) (p V q) & (p V r) → (q V r) is a tautology (i.e. always true)
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  2. #2
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    Hello, captainjapan!


    (a \to b) \;\equiv\: \sim a \vee b . I call it ADI (alternate definition of implication).



    (a)\;\;[\sim p \to (q \to r)] \: \equiv\: [q \to (p \vee r)]
    On the left side we have:

    . . \begin{array}{ccc}\sim p \to (q \to r) & & \text{Given} \\ \\ [-4mm]<br /> <br />
p \vee (q \to r) & & \text{ADI} \\ \\ [-4mm]<br /> <br />
p \vee (\sim q \vee r) & & \text{ADI} \\ \\ [-4mm]<br /> <br />
\sim q \vee (p \vee r) & & \text{comm., assoc.} \\ \\[-4mm]<br /> <br />
q \to (p \vee r) & & \text{ADI} \end{array}



    (b)\;\;\sim (p \to \; \sim q)\: \wedge  \sim(p \vee q) is a contradiction (i.e. always false).

    . . \begin{array}{ccc}<br />
\sim(p \to \:\sim q)\; \wedge \sim(p \vee q) & & \text{Given} \\ \\[-4mm]<br />
\sim(\sim p \:\vee \sim q) \;\wedge \sim(p \vee q) & & \text{ADI} \\ \\[-4mm]<br />
(p \wedge q) \wedge (\sim p \:\wedge \sim q) & & \text{DeMorgan} \\ \\[-4mm]<br />
(p \:\wedge \sim p) \wedge (q \:\wedge \sim q) & & \text{comm, assoc.} \\<br />
f \wedge f \\ \\[-4mm]<br />
f<br />
\end{array}

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  3. #3
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    Quote Originally Posted by Soroban View Post
    (a \to b) \;\equiv\: \sim a \vee b . I call it ADI (alternate definition of implication).
    FYI: In formal logic it is called Material Implication (Impl)
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