Question

The statement \( [(p \rightarrow q) \wedge \sim q] \rightarrow r \) is a tautology when \( r \) is equivalent to:

• A. \( p \wedge \sim q \)
• B. \( q \vee p \)
• C. \( p \wedge q \)
• D. \( \sim q \)

Hint:

For tautologies, ensure that the statement holds true under all possible truth values for the components. Analyze each component and simplify to identify equivalent expressions.

Answer 1

The Correct Option is D

A tautology is a statement that is always true, regardless of the truth values of its components. The statement \( [(p \rightarrow q) \wedge \sim q] \rightarrow r \) will be a tautology if \( r \) is equivalent to \( \sim q \). Let's analyze the antecedent: \( (p \rightarrow q) \wedge \sim q \). This is only true if \( p \rightarrow q \) is true and \( \sim q \) is true. If \( \sim q \) is true, then \( q \) must be false. If \( q \) is false and \( p \rightarrow q \) is true, then \( p \) must also be false (because if \( p \) were true, \( p \rightarrow q \) would be false). Thus, the antecedent is only true when both \( p \) and \( q \) are false. The entire statement \( [(p \rightarrow q) \wedge \sim q] \rightarrow r \) is a conditional statement which is only false when the antecedent is true and the consequent is false. To make it a tautology, \( r \) must be true whenever \( (p \rightarrow q) \wedge \sim q \) is true, which is equivalent to \( \sim q \) being true. Therefore, \( r \) must be equivalent to \( \sim q \).