Question

Which of the following statements is logically equivalent to \(\sim (p\leftrightarrow q)\) ?

• A. \(\sim p\rightarrow q\)
• B. \(\sim p\leftrightarrow \sim q\)
• C. \(\sim (q\rightarrow \sim p)\)
• D. \(p\leftrightarrow \sim q\)

Hint:

Negating a biconditional gives true when p and q differ. Compare truth tables with the options.

Answer 1

The Correct Option is D

Step 1: Meaning of the biconditional
\(p\leftrightarrow q\) is true when \(p\) and \(q\) have the same truth value. So \(\sim(p\leftrightarrow q)\) is true when they differ.

Step 2: Truth table
For (p,q) = (T,T): \(p\leftrightarrow q\) = T, so \(\sim\) gives F. For (T,F): F, so \(\sim\) gives T. For (F,T): F, so \(\sim\) gives T. For (F,F): T, so \(\sim\) gives F. Thus \(\sim(p\leftrightarrow q)\) has column F, T, T, F.

Step 3: Check option D
\(p\leftrightarrow\sim q\): (T,T): \(\sim q\)=F, so F. (T,F): \(\sim q\)=T, so T. (F,T): \(\sim q\)=F, so T. (F,F): \(\sim q\)=T, so F. Column F, T, T, F, which matches.

Step 4: Check the others
(A) \(\sim p\rightarrow q\) is \(p\vee q\), column T, T, T, F. No. (B) \(\sim p\leftrightarrow\sim q\) equals \(p\leftrightarrow q\), column T, F, F, T. No. (C) \(\sim(q\rightarrow\sim p)\) equals \(q\wedge p\), column T, F, F, F. No.

Final Answer:
\(\sim(p\leftrightarrow q)\equiv p\leftrightarrow\sim q\), option (D). \[ \boxed{p\leftrightarrow\sim q} \]