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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Here's the Solution to this Question

Since

\neg \left( {P \leftrightarrow Q} \right) \equiv \neg \left( {\left( {P \to Q} \right) \wedge \left( {Q \to P} \right)} \right) \equiv \neg \left( {\left( {\neg P \vee Q} \right) \wedge \left( {\neg Q \vee P} \right)} \right) \equiv \neg \left( {\neg P \vee Q} \right) \vee \neg \left( {\neg Q \vee P} \right) \equiv \left( {P \wedge \neg Q} \right) \vee \left( {Q \wedge \neg P} \right)

Then

\neg \left( {P \leftrightarrow Q} \right) \leftrightarrow \left( {P \wedge \neg Q} \right) \vee \left( {\neg P \wedge Q} \right)

Since

\left( {P \wedge \neg Q} \right) \vee \left( {\neg P \wedge Q} \right) \equiv \left( {P \vee \neg P} \right) \wedge \left( {\neg Q \vee \neg P} \right) \wedge \left( {P \vee Q} \right) \wedge \left( {\neg Q \vee Q} \right) \equiv 1 \wedge \left( {\neg Q \vee \neg P} \right) \wedge \left( {P \vee Q} \right) \wedge 1 \equiv \left( {\neg Q \vee \neg P} \right) \wedge \left( {P \vee Q} \right) \equiv \left( {P \vee Q} \right) \wedge \neg \left( {P \wedge Q} \right)

Then

\left( {P \wedge \neg Q} \right) \vee \left( {\neg P \wedge Q} \right) \leftrightarrow \left( {P \vee Q} \right) \wedge \neg \left( {P \wedge Q} \right)

But then

\neg \left( {P \leftrightarrow Q} \right) \leftrightarrow \left( {P \vee Q} \right) \wedge \neg \left( {P \wedge Q} \right) \leftrightarrow \left( {P \wedge \neg Q} \right) \vee \left( {\neg P \wedge Q} \right)

Q. E. D.


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Question ID: mtid-5-stid-8-sqid-2775-qpid-1332