The compound statement $$(P \vee Q) \wedge (\sim P) \Rightarrow Q$$ equivalent to:

We have the compound statement

$$\bigl((P \vee Q) \wedge (\sim P)\bigr)\;\Rightarrow\;Q.$$

First we recall the basic logical equivalence for an implication:

$$A \Rightarrow B \;\equiv\; \sim A \;\vee\; B.$$

Here,

$$A = (P \vee Q)\wedge(\sim P) \quad\text{and}\quad B = Q.$$

Applying the rule we obtain

$$\bigl((P \vee Q)\wedge(\sim P)\bigr)\Rightarrow Q \;\equiv\; \sim\bigl((P \vee Q)\wedge(\sim P)\bigr)\;\vee\;Q.$$

Now we remove the negation inside by using De Morgan’s law:

$$\sim(X\wedge Y)\;\equiv\; \sim X\;\vee\;\sim Y.$$

So, with

$$X = (P \vee Q)\quad\text{and}\quad Y = (\sim P),$$

we have

$$\sim\bigl((P \vee Q)\wedge(\sim P)\bigr) = \bigl(\sim(P \vee Q)\bigr)\;\vee\;\bigl(\sim(\sim P)\bigr) = (\sim P\wedge\sim Q)\;\vee\;P.$$

The expression therefore becomes

$$\bigl((\sim P\wedge\sim Q)\;\vee\;P\bigr)\;\vee\;Q.$$

The associative and commutative laws of $$\vee$$ allow us to regroup and reorder the disjunction:

$$ \bigl((\sim P\wedge\sim Q)\;\vee\;P\bigr)\;\vee\;Q \;=\; P\;\vee\;Q\;\vee\;(\sim P\wedge\sim Q). $$

Next we distribute $$\vee$$ over $$\wedge$$ to simplify the mixed term:

$$ P\;\vee\;Q\;\vee\;(\sim P\wedge\sim Q) = (P\;\vee\;Q)\;\vee\;(\sim P\wedge\sim Q) = \bigl(P\;\vee\;Q\;\vee\;\sim P\bigr)\;\wedge\;\bigl(P\;\vee\;Q\;\vee\;\sim Q\bigr). $$

Each of the two factors contains a full pair of complementary literals:

$$P\;\vee\;\sim P = \text{T},\quad Q\;\vee\;\sim Q = \text{T}.$$

Hence both brackets reduce to T (truth), and we are left with

$$\text{T}\;\wedge\;\text{T} = \text{T}.$$

Thus the given compound statement is a tautology; it is always true for every possible truth-value assignment to $$P$$ and $$Q$$.

Among the provided options, Option D is

$$\sim(P \Rightarrow Q)\;\Leftrightarrow\;P\wedge\sim Q,$$

which is itself a biconditional connecting two logically equivalent expressions (since $$\sim(P \Rightarrow Q)\equiv P\wedge\sim Q$$). A biconditional between two identical propositions is also a tautology, i.e. it is always true. Therefore the original statement and Option D are equivalent.

Hence, the correct answer is Option 4.