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

We are given the Boolean expression $$\sim(p \vee q) \vee (\sim p \wedge q)$$ and we must find an equivalent, simpler form.

First recall De Morgan’s law, which states that $$\sim (A \vee B)=\sim A \wedge \sim B.$$

Applying this law to the first part, we have:

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

Substituting this result into the original expression, we get

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

Now observe that $$\sim p$$ is common in both terms of the disjunction. Using the distributive law $$A\wedge B \;\; \vee \;\; A\wedge C = A\wedge (B\vee C),$$ we factor out $$\sim p$$:

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

Inside the parentheses we have $$\sim q \vee q,$$ which is always true (a tautology). Denote the Boolean constant “true” by $$T$$. Hence

$$\sim q \vee q = T.$$

Substituting this back, we obtain

$$\sim p \wedge T.$$

In Boolean algebra, any proposition ANDed with $$T$$ remains unchanged, so

$$\sim p \wedge T = \sim p.$$

Thus, the original expression simplifies completely to $$\sim p.$$

Looking at the given options, $$\sim p$$ corresponds to Option B.

Hence, the correct answer is Option B.