The Boolean expression $$((p \wedge q) \vee (p \vee \sim q)) \wedge (\sim p \wedge \sim q)$$ is equivalent to

We have to simplify the Boolean expression $$\bigl((p \wedge q) \vee (p \vee \sim q)\bigr) \wedge (\sim p \wedge \sim q).$$ Throughout, we shall use the well-known Boolean identities: (i) Absorption Law $$X \vee (X \wedge Y)=X,$$ (ii) Distributive Law $$A \wedge (B \vee C)= (A \wedge B)\; \vee\; (A \wedge C),$$ (iii) Idempotent Law $$Z \wedge Z=Z,$$ and (iv) the fact that any statement conjoined with its negation is false, $$R \wedge (\sim R)=0.$$

First concentrate on the part $$ (p \wedge q) \vee (p \vee \sim q).$$ In order to apply the absorption law, we re-express $$(p \wedge q)$$ in a form that visibly contains $$(p \vee \sim q).$$ Observe that

$$ (p \vee \sim q)\;\wedge\; q \;=\; (p \wedge q)\; \vee\; (\sim q \wedge q).$$

Because $$\sim q \wedge q = 0,$$ the right-hand side reduces to $$p \wedge q.$$ Thus we have shown

$$p \wedge q \;=\; (p \vee \sim q)\;\wedge\; q.$$

Letting $$X = (p \vee \sim q) \quad\text{and}\quad Y = q,$$ we may rewrite the first bracket as $$ (X \wedge Y) \vee X.$$ Now, by the absorption law, $$ (X \wedge Y) \vee X = X.$$ Therefore

$$ (p \wedge q) \vee (p \vee \sim q) = p \vee \sim q.$$

Substituting this back, the whole expression becomes

$$ (p \vee \sim q)\; \wedge\; (\sim p \wedge \sim q).$$

Using associativity and commutativity of $$\wedge,$$ we group as

$$ (\sim p \wedge \sim q)\; \wedge\; (p \vee \sim q).$$

Now apply the distributive law with $$A = (\sim p \wedge \sim q),\; B = p,\; C = \sim q:$$

$$ (\sim p \wedge \sim q \wedge p)\; \vee\; (\sim p \wedge \sim q \wedge \sim q).$$

Simplify each part step by step. In the first term, $$\sim p \wedge p = 0,$$ so

$$\sim p \wedge \sim q \wedge p = 0 \wedge \sim q = 0.$$

In the second term, the idempotent law gives $$\sim q \wedge \sim q = \sim q,$$ therefore

$$\sim p \wedge \sim q \wedge \sim q = \sim p \wedge \sim q.$$

Hence the entire expression reduces to

$$ 0 \;\vee\; (\sim p \wedge \sim q) = \sim p \wedge \sim q.$$

Thus the simplified (and therefore equivalent) Boolean expression is $$ (\sim p) \wedge (\sim q).$$ This matches Option B.

Hence, the correct answer is Option B.