Question

If the truth value of the compound statement \([(p\leftrightarrow q)∧(q\rightarrow r)∧\sim r]\rightarrow (p∧\sim q)\) is false, then the truth values of the statement patterns \((p\rightarrow q)\leftrightarrow (q\rightarrow r)\) and \(\sim (p∨r)\rightarrow (q∧p)\) are, respectively ...

• A. \((T,T)\)
• B. \((T,F)\)
• C. \((F,T)\)
• D. \((F,F)\)

Hint:

A conditional is false only when the antecedent is true and the consequent is false.

Answer 1

The Correct Option is B

Step 1: Find the truth values of p, q, r
The given implication is false, so the antecedent is true and \(p\wedge\sim q\) is false. The antecedent is true only if \(\sim r\) is true, so \(r = F\).

Step 2: Use the other parts
\(q\rightarrow r\) must be true with \(r=F\), so \(q = F\). Then \(p\leftrightarrow q\) is true only if \(p = F\). Check: \(p\wedge\sim q = F\wedge T = F\), as required. So \(p=q=r=F\).

Step 3: First pattern
\((p\rightarrow q) = T\) and \((q\rightarrow r) = T\). So \(T\leftrightarrow T = T\).

Step 4: Second pattern
\(p\vee r = F\), so \(\sim(p\vee r) = T\). The consequent \(q\wedge p = F\). \(T\rightarrow F = F\).

Step 5: Result
The pair is \((T,F)\), option (B).

Final Answer:
The truth values are T and F. \[ \boxed{\text{(B)}\ (T,F)} \]