Question

The statement $\sim (p\leftrightarrow \sim q)$ is:

• A. equivalent to $p\leftrightarrow q$
• B. equivalent to $\sim p\leftrightarrow q$
• C. a tautology
• D. a fallacy

Answer 1

The correct option is A equivalent to $p\leftrightarrow q$

As we know,
$p\leftrightarrow q=(p\wedge q)\vee (\sim p\wedge \sim q)$

$p\leftrightarrow \sim q=(p\to \sim q)\wedge (\sim q\to p)$
$=(\sim p\vee \sim q)\wedge (q\vee p)$

Therefore, $\sim (p\leftrightarrow \sim q)$
$=\sim \left[\right(\sim p\vee \sim q)\wedge (q\vee p\left)\right]$
$=(p\wedge q)\vee (\sim p\wedge \sim q)$
$=p\leftrightarrow q$