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Joint Entrance Examination
Mathematics
Mathematical Reasoning
Question 64
Question 64 of 103
Medium
MCQ
The compound statement \[\left( { \sim (P \wedge Q)} \right) \vee \left( {( \sim P) \wedge Q} \right) \Rightarrow \left( {( \sim P) \wedge ( \sim Q)} \right)\] is equivalent to
A
\[P \vee Q\]
B
\[P \wedge \sim Q\]
C
\[ \sim (P \Rightarrow Q)\]
D
\[ \sim (P \Rightarrow Q) \Leftrightarrow P \wedge \sim Q\]
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Official solution
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\begin{aligned} &(\sim(P \wedge Q)) \vee((\sim P) \wedge Q) \Rightarrow((\sim P) \wedge(\sim Q)) \\\\ &(\sim P \vee \sim Q) \vee(\sim P \wedge Q) \Rightarrow(\sim P \wedge \sim Q) \\\\ & \Rightarrow(\sim P \vee \sim Q \vee \sim P) \wedge(\sim P \vee \sim Q \vee Q) \Rightarrow(\sim P \wedge \sim Q) \\\\ & \Rightarrow(\sim P \vee \sim Q) \wedge(T) \Rightarrow(\sim P \wedge \sim Q) \\\\ & \Rightarrow(\sim P \vee \sim Q) \Rightarrow(\sim P \wedge \sim Q) \\\\ & \Rightarrow \sim(\sim P \vee \sim Q) \vee(\sim P \wedge \sim Q) \\\\ & \Rightarrow(P \wedge Q) \vee(\sim P \wedge \sim Q) \Rightarrow(\sim P \vee Q) \wedge(\bullet Q \vee P) \end{aligned}
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