JEE 2021 — Mathematics PYQ

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Question

JEE 2021 — Mathematics PYQ

JEE | Mathematics | 2021

The compound statement $(\sim(P \wedge Q)) \vee ((\sim P) \wedge Q) \Rightarrow ((\sim P) \wedge (\sim Q))$ is equivalent to

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Explanation

Step 1: Pehle part ko simplify karein $(\sim(P \wedge Q))$

De Morgan's law ke hisaab se:

\sim(P \wedge Q) \equiv (\sim P \vee \sim Q)

Step 2: Doosre part ko simplify karein $((\sim P) \wedge Q) \Rightarrow ((\sim P) \wedge (\sim Q))$

Hume pata hai ki $A \Rightarrow B \equiv \sim A \vee B$ . Yahan $A = ((\sim P) \wedge Q)$ aur $B = ((\sim P) \wedge (\sim Q))$ hai.

\sim((\sim P) \wedge Q) \vee ((\sim P) \wedge (\sim Q))

\equiv (P \vee \sim Q) \vee ((\sim P) \wedge (\sim Q))

Distributive law use karne par:

\equiv (P \vee \sim Q \vee \sim P) \wedge (P \vee \sim Q \vee \sim Q)

Kyunki $(P \vee \sim P)$ hamesha True ( $T$ ) hota hai:

\equiv (T \vee \sim Q) \wedge (P \vee \sim Q)

\equiv T \wedge (P \vee \sim Q) \equiv (P \vee \sim Q)

Step 3: Dono parts ko $\vee$ (OR) ke saath jodein:

Ab hamare paas bacha:

(\sim P \vee \sim Q) \vee (P \vee \sim Q)

Ise rearrange karne par:

(\sim P \vee P) \vee (\sim Q \vee \sim Q)

T \vee \sim Q

Choose the correct answer:

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