A tautology is a statement invariably true, irrespective of the truth values of its constituent parts.
The statement \( [(p \rightarrow q) \wedge \sim q] \rightarrow r \) is a tautology if \( r \) is equivalent to \( \sim q \).
Analyzing the antecedent: \( (p \rightarrow q) \wedge \sim q \). This holds true only when \( p \rightarrow q \) is true and \( \sim q \) is true. If \( \sim q \) is true, then \( q \) is false. With \( q \) false and \( p \rightarrow q \) true, \( p \) must also be false (as a true \( p \) would render \( p \rightarrow q \) false). Consequently, the antecedent is true solely when both \( p \) and \( q \) are false.
The complete statement \( [(p \rightarrow q) \wedge \sim q] \rightarrow r \) is a conditional statement that is false only when its antecedent is true and its consequent is false. For this to be a tautology, \( r \) must be true whenever \( (p \rightarrow q) \wedge \sim q \) is true, which is synonymous with \( \sim q \) being true. Thus, \( r \) must be equivalent to \( \sim q \).