Question

If \(p\rightarrow (q∨\sim r)\) is false, then the truth values of \((p\leftrightarrow q)∧r\) and \(\sim p\rightarrow \sim q\) are :

• 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 false.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
\(p\to(q\vee\sim r)\) is false only when \(p\) is T and \(q\vee\sim r\) is F.

Step 2: Key Formula or Approach
\(q\vee\sim r\) is false only when both \(q\) and \(\sim r\) are false, so \(q=F\) and \(r=T\).

Step 3: Detailed Explanation
So \(p=T\), \(q=F\), \(r=T\).
\((p\leftrightarrow q)\wedge r\): \(p\leftrightarrow q=T\leftrightarrow F=F\). \(F\wedge T=F\).
\(\sim p\to\sim q\): \(F\to T=T\).
The truth values are \((F,T)\).

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