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यदि $P$ तथा $Q$ दो कथन हैं, तो निम्न में से कौन-सा मिश्र कथन पुनरूक्ति है ?
$(( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow Q$
$(( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow \sim P$
$(( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow P$
$(( P \Rightarrow Q ) \wedge \sim Q ) \Rightarrow( P \wedge Q )$
Solution
$LHS$ of all the options are some i.e.
$((P \rightarrow Q) \wedge \sim Q)$
$\equiv(\sim P \vee Q ) \wedge \sim Q$
$\equiv(\sim P \wedge \sim Q ) \vee( Q \wedge \sim Q )$
$\equiv \sim P \wedge \sim Q$
$(A)$ $(\sim P \wedge \sim Q) \rightarrow Q$
$\equiv \sim(\sim P \wedge \sim Q) \vee Q$
$\equiv(P \vee Q) \vee Q \neq$ tautology
$(B)$ $(\sim P \wedge \sim Q) \rightarrow \sim P$
$\equiv \sim(\sim P \wedge \sim Q ) \vee \sim P$
$\equiv( P \vee Q ) \vee \sim P$
$(C)$ $(\sim P \wedge \sim Q) \rightarrow P$
$\equiv(P \vee Q) \vee P \neq$ Tautology
$(D)\,(\sim P \wedge \sim Q) \rightarrow(P \wedge Q)$
$\equiv(P \vee Q) \vee(P \wedge Q) \neq$ Tautology
Aliter:
| $P$ | $Q$ | $P \vee Q$ | $P \vee Q$ | $\sim P$ | $(P \vee Q) \vee \sim P$ |
| $T$ | $T$ | $T$ | $T$ | $F$ | $T$ |
| $T$ | $F$ | $T$ | $F$ | $F$ | $T$ |
| $F$ | $T$ | $T$ | $F$ | $T$ | $T$ |
| $F$ | $F$ | $F$ | $F$ | $T$ | $T$ |
