Question

The statement pattern \((p∧q)\rightarrow (r∨\sim s)\) is false. Then the truth values of \(p,q,r\) and \(s\) are respectively

• A. T, F, T, T
• B. T, T, F, T
• C. T, T, T, F
• D. T, T, F, F

Hint:

An implication is false only when the first part is true and the second part is false.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
An implication \(A\rightarrow B\) is false in exactly one case: \(A\) is true and \(B\) is false. All other combinations make it true.

Step 2: Apply to the statement:
Here \(A=(p\wedge q)\) and \(B=(r\vee\sim s)\). So \(p\wedge q\) must be true and \(r\vee\sim s\) must be false.

Step 3: Find p and q:
A conjunction is true only when both parts are true. So \(p=T\) and \(q=T\).

Step 4: Find r and s:
A disjunction is false only when both parts are false. So \(r=F\) and \(\sim s=F\), which means \(s=T\).

Step 5: Match the options:
The values are \(p,q,r,s=T,T,F,T\). Option (A) has \(q=F\), so \(p\wedge q\) is false. Option (C) has \(r=T\), so \(B\) is true. Option (D) has \(s=F\), so \(\sim s=T\) and \(B\) is true. In all three the whole statement would be true.

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