Question

If the compound statement \((p\wedge \sim q)\rightarrow (q\vee \sim r)\) is false then which of the following statements is a true ?

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

Hint:

A conditional is false only when the first part is true and the second part is false. Use this to fix p, q and r.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(A\rightarrow B\) is false only when \(A\) is true and \(B\) is false.

Step 2: Fix the Truth Values
\(p\wedge\sim q\) is true, so \(p=T\) and \(q=F\).
\(q\vee\sim r\) is false, so \(q=F\) and \(\sim r=F\), which gives \(r=T\).

Step 3: Check Option (A)
\((p\wedge r)\rightarrow q\) is \(T\rightarrow F\), which is false.

Step 4: Check Option (B)
\(\sim p\vee q\) is \(F\vee F=F\). Then \(F\rightarrow r\) is true.

Step 5: Check Options (C) and (D)
\((p\rightarrow q)\) is \(F\), so (C) is false. \(q\wedge r=F\), so \(p\rightarrow(q\wedge r)\) is false.

Final Answer:
Only the statement (not p or q) implies r is true. \[ \boxed{\text{(B) }(\sim p\vee q)\rightarrow r} \]