Which of the following statements is a tautology?

We need to determine which of the given statements is a tautology (always true regardless of truth values of $$ p $$ and $$ q $$).

Recall that $$ A \Rightarrow B $$ is equivalent to $$ \sim A \lor B $$.

$$q \Rightarrow (\sim p \lor q) \equiv \sim q \lor (\sim p \lor q) \equiv \sim p \lor (\sim q \lor q) \equiv \sim p \lor T \equiv T$$

Since $$ \sim q \lor q $$ is always true (law of excluded middle), the entire expression simplifies to True. This is a tautology.

Option A: $$ (\sim p \lor q) \Rightarrow p \equiv \sim(\sim p \lor q) \lor p \equiv (p \land \sim q) \lor p \equiv p $$. This is not a tautology (false when $$ p = F $$).

Option B: $$ p \Rightarrow (\sim p \lor q) \equiv \sim p \lor (\sim p \lor q) \equiv \sim p \lor q $$. This is not a tautology (false when $$ p = T, q = F $$).

Option C: $$ (\sim p \lor q) \Rightarrow q \equiv \sim(\sim p \lor q) \lor q \equiv (p \land \sim q) \lor q \equiv (p \lor q) \land (\sim q \lor q) \equiv p \lor q $$. This is not a tautology (false when $$ p = F, q = F $$).

The tautology is $$ q \Rightarrow (\sim p \lor q) $$, which corresponds to Option D.