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Solution to Show that ¬ (P\iff⟺Q)\iff⟺(P V Q) Λ ¬(P Λ Q) \iff⟺(P Λ ¬Q) V (¬ … - Sikademy
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Archangel Macsika

Show that ¬ (P\iff⟺Q)\iff⟺(P V Q) Λ ¬(P Λ Q) \iff⟺(P Λ ¬Q) V (¬ P Λ Q) without using truth table

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Solution:

\begin{aligned} &\neg(P \Leftrightarrow Q) \\ &\Leftrightarrow \neg((P \rightarrow Q) \wedge(Q \rightarrow P)) \\ &\Leftrightarrow \neg((P \vee \neg Q) \wedge(Q \vee \neg P)) \\ &\Leftrightarrow \neg(((P \vee \neg Q) \wedge Q) \vee((P \vee \neg Q) \wedge \neg P)) \\ &\Leftrightarrow \neg((P \wedge Q) \vee(\neg Q \wedge Q) \vee((P \wedge \neg P) \vee(\neg Q \wedge \neg P)) \\ &\Leftrightarrow \neg((P \wedge Q) \vee \neg(P \vee Q)) \\ &\Leftrightarrow \neg(P \wedge Q) \wedge \neg(\neg(P \vee Q)) \\ &\Leftrightarrow(P \vee Q) \wedge \neg(P \wedge Q) \ldots(1) \end{aligned}

Also,

\begin{aligned} &\neg(P \Leftrightarrow Q) \\ &\Leftrightarrow \neg((P \rightarrow Q) \wedge(Q \rightarrow P)) \\ &\Leftrightarrow \neg((P \vee \neg Q) \wedge(Q \vee \neg P)) \\ &\Leftrightarrow(\neg P \wedge Q) \vee(\neg Q \wedge P) \ldots(2) \end{aligned}

Therefore the result follows from (1) and (2).

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Question ID: mtid-5-stid-8-sqid-2647-qpid-1117