Question

Statement - 1: $\sim (p\leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q$.
Statement - 2: $\sim (p\leftrightarrow \sim q)$ is a tautology.

• A. Statement - 1 is true, Statement - 2 is true;
  Statement - 2 is not a correct explanation for statement - 1.
• B. Statement - 1 is true, Statement - 2 is false.
• C. Statement - 1 is false, Statement - 2 is true.
• D. Statement - 1 is true, Statement - 2 is true,
  Statement - 2 is a correct explanation for statement - 1.

Answer 1

The correct option is B Statement - 1 is true, Statement - 2 is false.

The truth table for the logical statements, involved in statement 1, is as follows:
$\begin{array}{cccccc}p& q& \sim q& p\leftrightarrow \sim q& \sim (p\leftrightarrow \sim q)& p\leftrightarrow q\\ T& T& F& F& T& T\\ T& F& T& T& F& F\\ F& T& F& T& F& F\\ F& F& T& F& T& T\end{array}$
We observe the columns for $\sim (p\leftrightarrow \sim q)$ and $p\leftrightarrow q$ are identical, therefore
$\sim (p\leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q$
But $\sim (p\leftrightarrow \sim q)$ is not a tautology as all entries in its column are not T.
$\therefore$ Statement - 1 is true but statement - 2 is false.