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

The logical equivalence of the biconditional statement is given by:

$p \leftrightarrow q \equiv (p \rightarrow q) \wedge (q \rightarrow p)$

Taking the negation of both sides:

$\sim(p \leftrightarrow q) \equiv \sim((p \rightarrow q) \wedge (q \rightarrow p))$

Using De Morgan's laws:

$\equiv \sim(p \rightarrow q) \vee \sim(q \rightarrow p)$

Since $p \rightarrow q \equiv \sim p \vee q$, we get:

$\equiv \sim(\sim p \vee q) \vee \sim(\sim q \vee p)$

Applying De Morgan's laws again:

$\equiv (p \wedge \sim q) \vee (q \wedge \sim p)$

Now, evaluating the expression $p \leftrightarrow \sim q$:

$p \leftrightarrow \sim q \equiv (p \rightarrow \sim q) \wedge (\sim q \rightarrow p)$

$\equiv (\sim p \vee \sim q) \wedge (q \vee p)$

Using the distributive law:

$\equiv (\sim p \wedge q) \vee (\sim p \wedge p) \vee (\sim q \wedge q) \vee (\sim q \wedge p)$

Since $\sim p \wedge p \equiv F$ and $\sim q \wedge q \equiv F$:

$\equiv (\sim p \wedge q) \vee (p \wedge \sim q)$

Comparing the two results, we conclude that $\sim(p \leftrightarrow q) \equiv p \leftrightarrow \sim q$.

Answer: $p \leftrightarrow \sim q$