Question

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

• A. ((P$\Rightarrow$Q) $\wedge$ $\neg$Q) $\Rightarrow$ P
• B. ((P$\Rightarrow$Q) $\wedge$ $\neg$Q) $\Rightarrow$ Q
• C. ((P$\Rightarrow$Q) $\wedge$ $\neg$Q) $\Rightarrow$ $\neg$P
• D. ((P$\Rightarrow$Q) $\wedge$ $\neg$Q) $\Rightarrow$ (P $\wedge$ Q)

Hint:

This is the rule of {Modus Tollens}: If $P$ implies $Q$, and $Q$ is false, then $P$ must be false.

Answer 1

The Correct Option is C

Step 1: Let's simplify the antecedent: $(P \Rightarrow Q) \wedge \neg Q$. $(P \Rightarrow Q)$ is equivalent to $(\neg P \vee Q)$. So, $(\neg P \vee Q) \wedge \neg Q \equiv (\neg P \wedge \neg Q) \vee (Q \wedge \neg Q) \equiv (\neg P \wedge \neg Q) \vee F \equiv (\neg P \wedge \neg Q)$.
Step 2: Now check option (C): $(\neg P \wedge \neg Q) \Rightarrow \neg P$. Since $(A \wedge B) \Rightarrow A$ is always a tautology, $(\neg P \wedge \neg Q) \Rightarrow \neg P$ is a tautology.