The negation of $$(p \wedge (-q)) \vee (-p)$$ is equivalent to

We need to find the negation of $$(p \wedge (\neg q)) \vee (\neg p)$$.

Simplify the given expression.

Using the distributive law, we expand:

$$(p \wedge (\neg q)) \vee (\neg p) = (\neg p \vee p) \wedge (\neg p \vee \neg q)$$

Since $$\neg p \vee p$$ is a tautology (always true):

$$= T \wedge (\neg p \vee \neg q) = \neg p \vee \neg q$$

Apply De Morgan's Law.

By De Morgan's Law, $$\neg p \vee \neg q = \neg(p \wedge q)$$. So:

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

Find the negation.

The negation of $$\neg(p \wedge q)$$ is:

$$\neg[\neg(p \wedge q)] = p \wedge q$$

Verification using truth table:

When $$p = T, q = T$$: Original = $$(T \wedge F) \vee F = F$$. Negation = $$T$$. And $$p \wedge q = T$$. ✔

When $$p = T, q = F$$: Original = $$(T \wedge T) \vee F = T$$. Negation = $$F$$. And $$p \wedge q = F$$. ✔

When $$p = F, q = T$$: Original = $$(F \wedge F) \vee T = T$$. Negation = $$F$$. And $$p \wedge q = F$$. ✔

When $$p = F, q = F$$: Original = $$(F \wedge T) \vee T = T$$. Negation = $$F$$. And $$p \wedge q = F$$. ✔

The negation matches $$p \wedge q$$ in all cases.

The answer is Option B: $$p \wedge q$$.