Question

The statement pattern $[(p \rightarrow q)\land \sim q] \rightarrow r$ is a tautology when $r$ is equivalent to

• A. $\text{p}\land \sim \text{q}$
• B. $q \lor p$
• C. $p \land q$
• D. $\sim q$

Hint:

Simplify the antecedent first, then find which r makes the implication true for every truth value of p and q.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
An implication \(A\rightarrow r\) is a tautology if r is true whenever A is true.

Step 2: Simplify the antecedent
\((p\rightarrow q)\land\sim q\equiv(\sim p\lor q)\land\sim q\equiv\sim p\land\sim q\).

Step 3: Required r
The antecedent is true only when both p and q are false. So r must be true when p and q are both false.

Step 4: Test the options
\(p\land\sim q\), \(q\lor p\) and \(p\land q\) are all false when p and q are false, while \(\sim q\) is true there.

Step 5: Compare the options
Hence \(r\equiv\sim q\), the last option. Also \(\sim p\land\sim q\rightarrow\sim q\) is always true.

Final Answer:
r must be equivalent to \(\sim q\). \[ \boxed{\text{(D) }\sim q} \]