Question

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

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

Answer 1

The correct option is B $((P\Rightarrow Q)\wedge \sim Q)\Rightarrow \sim P$

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

Now,
$\left(1\right)\sim (P\vee Q)\Rightarrow P$
$\equiv (P\vee Q)\vee P$
$\equiv P\vee Q$

$\left(2\right)\sim (P\vee Q)\Rightarrow \sim P$
$\equiv (P\vee Q)\vee \sim P$
$\equiv T$

$\left(3\right)\sim (P\vee Q)\Rightarrow (P\wedge Q)$
$\equiv (P\vee Q)\vee (P\wedge Q)$
$\equiv P\vee Q$

$\left(4\right)\sim (P\vee Q)\Rightarrow Q$
$\equiv (P\vee Q)\vee Q$
$\equiv P\vee Q$

Thus, $((P\Rightarrow Q)\wedge \sim Q)\Rightarrow \sim P$ is a tautology.