The proposition $$p \to \sim(p \wedge \sim q)$$ is equivalent to:

We have to simplify the proposition $$p \to \sim(p \wedge \sim q)$$ and then compare the final result with the four given options.

First recall the logical equivalence for an implication. The standard formula is:

$$a \to b \equiv (\sim a) \vee b.$$

In our problem the role of $$a$$ is played by $$p$$ and the role of $$b$$ is played by $$\sim(p \wedge \sim q).$$ Substituting into the formula we obtain

$$p \to \sim(p \wedge \sim q) \equiv (\sim p) \vee \bigl[\;\sim(p \wedge \sim q)\bigr].$$

Now we tackle the term $$\sim(p \wedge \sim q)$$. To simplify a negation of a conjunction, we use De Morgan’s law, which states:

$$\sim(A \wedge B) \equiv (\sim A) \vee (\sim B).$$

Here $$A$$ is $$p$$ and $$B$$ is $$\sim q$$. Applying the law carefully gives

$$\sim(p \wedge \sim q) \equiv (\sim p) \vee \bigl[\sim(\sim q)\bigr].$$

A double negation disappears, because $$\sim(\sim q) \equiv q$$. Hence the entire right-hand side becomes

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

We now substitute this back into the expression we obtained after the implication step:

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

The disjunction is associative and idempotent, meaning that repeating the same statement inside a string of ORs does not change the result. Explicitly,

$$ (\sim p) \vee (\sim p) \equiv \sim p, \quad\text{and}\quad (\sim p) \vee \bigl[(\sim p) \vee q\bigr] \equiv (\sim p) \vee q.$$

Therefore the original proposition simplifies all the way down to

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

This final expression matches exactly what is written in Option B. Hence, the correct answer is Option B.