Answer to Question #218792 in Discrete Mathematics for ratnakar

Question #218792

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


1
Expert's answer
2022-01-31T12:32:02-0500

Since

¬(PQ)¬((PQ)(QP))¬((¬PQ)(¬QP))¬(¬PQ)¬(¬QP)(P¬Q)(Q¬P)\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

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

Since

(P¬Q)(¬PQ)(P¬P)(¬Q¬P)(PQ)(¬QQ)1(¬Q¬P)(PQ)1(¬Q¬P)(PQ)(PQ)¬(PQ)\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

(P¬Q)(¬PQ)(PQ)¬(PQ)\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

¬(PQ)(PQ)¬(PQ)(P¬Q)(¬PQ)\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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