If the truth value of the statement $$(P \wedge (\sim R)) \to ((\sim R) \wedge Q)$$ is F, then the truth value of which of the following is F?

The statement $$(P \wedge (\sim R)) \to ((\sim R) \wedge Q)$$ is given to be False. An implication $$p \to q$$ is false only when $$p$$ is True and $$q$$ is False, so we require $$P \wedge (\sim R) = T$$ and $$(\sim R) \wedge Q = F$$. From $$P \wedge (\sim R) = T$$ we deduce $$P = T$$ and $$\sim R = T$$, hence $$R = F$$. Since $$(\sim R) \wedge Q = F$$ and $$\sim R = T$$, it follows that $$Q = F$$. Therefore $$P = T,\ Q = F,\ R = F$$.

Substituting these truth values into each option, we find:

Option A: $$P \vee Q \to \sim R = (T \vee F) \to T = T \to T = T$$

Option B: $$R \vee Q \to \sim P = (F \vee F) \to F = F \to F = T$$

Option C: $$\sim(P \vee Q) \to \sim R = \sim(T) \to T = F \to T = T$$

Option D: $$\sim(R \vee Q) \to \sim P = \sim(F) \to F = T \to F = F$$

The statement with truth value False is $$\boxed{\sim(R \vee Q) \to \sim P}$$, so the answer is Option D.