Question

If the statement pattern \((p\vee \sim q)\wedge (\sim p\vee q)\) is true, then which of the following is true?

• A. \(p=T,q=F\)
• B. both \(p\) and \(q\) have the same truth value
• C. \(p\rightarrow \sim q\) is a contradiction
• D. \(p=F,q=T\)

Hint:

Recognise the pattern as a pair of converse implications, which together form a biconditional.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The pattern \((p\vee \sim q)\wedge(\sim p\vee q)\) is the conjunction of two implications.

Step 2: Rewrite the pattern
\(p\vee \sim q\equiv q\rightarrow p\) and \(\sim p\vee q\equiv p\rightarrow q\).
So the pattern is \((q\rightarrow p)\wedge(p\rightarrow q)\equiv p\leftrightarrow q\).

Step 3: Use the given condition
The pattern is true only when \(p\) and \(q\) have the same truth value (both \(T\) or both \(F\)).

Step 4: Compare the options
\(p=T,q=F\) and \(p=F,q=T\) give a false pattern. For \(p=q=T\), \(p\rightarrow \sim q\) is false, but for \(p=q=F\) it is true, so it is not a contradiction.
Only the statement that both have the same truth value is always true.

Final Answer:
If the pattern is true, then p and q must have the same truth value. \[ \boxed{\text{(B) }\text{both }p\text{ and }q\text{ have the same truth value}} \]