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

We need to check which compound statement is a tautology. Option 2 is $$((P \Rightarrow Q) \wedge \sim Q) \Rightarrow \sim P$$. This is precisely the law of Modus Tollens, a well-known logical tautology: if $$P$$ implies $$Q$$ and $$Q$$ is false, then $$P$$ must be false.

To verify, suppose the antecedent $$(P \Rightarrow Q) \wedge \sim Q$$ is true. Then $$\sim Q$$ is true, so $$Q$$ is false. Since $$P \Rightarrow Q$$ is also true and $$Q$$ is false, $$P$$ must be false (if $$P$$ were true, $$P \Rightarrow Q$$ would be false). Hence $$\sim P$$ is true, and the implication holds. If the antecedent is false, the implication is vacuously true. Therefore the statement is always true, confirming it is a tautology.