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(B) Let us simplify the antecedent $((P \Rightarrow Q) \wedge \sim Q)$ for all options:$((P \Rightarrow Q) \wedge \sim Q) \equiv ((\sim P \vee Q) \wed

Vedclass · Accepted Answer

$(( P$ $\Rightarrow Q ) \wedge \sim Q )$ $\Rightarrow \sim P$

Vedclass · Answer

(B) Let us simplify the antecedent $((P \Rightarrow Q) \wedge \sim Q)$ for all options:$((P \Rightarrow Q) \wedge \sim Q) \equiv ((\sim P \vee Q) \wedge \sim Q)$Using the distributive law: $(\sim P \wedge \sim Q) \vee (Q \wedge \sim Q)$Since $(Q \wedge \sim Q) \equiv F$,we have $(\sim P \wedge \sim Q) \vee F \equiv \sim P \wedge \sim Q$.Now,check each option:$(A) (\sim P \wedge \sim Q)$ $\Rightarrow Q \equiv \sim(\sim P \wedge \sim Q) \vee Q \equiv (P \vee Q) \vee Q \equiv P \vee Q$ (Not a tautology)$(B) (\sim P \wedge \sim Q)$ $\Rightarrow \sim P \equiv \sim(\sim P \wedge \sim Q) \vee \sim P \equiv (P \vee Q) \vee \sim P \equiv (P \vee \sim P) \vee Q \equiv T \vee Q \equiv T$ (Tautology)$(C) (\sim P \wedge \sim Q)$ $\Rightarrow P \equiv \sim(\sim P \wedge \sim Q) \vee P \equiv (P \vee Q) \vee P \equiv P \vee Q$ (Not a tautology)$(D) (\sim P \wedge \sim Q) \Rightarrow (P \wedge Q) \equiv (P \vee Q) \vee (P \wedge Q) \equiv P \vee Q$ (Not a tautology)Therefore,the correct option is $B$.

If $P$ and $Q$ are two statements,then which of the following compound statements is a tautology?

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