Let $$p$$ and $$q$$ be any two logical statements and $$r : p \rightarrow (\sim p \vee q)$$. If $$r$$ has a truth value $$F$$, then the truth values of $$p$$ and $$q$$ are respectively:

We are given that $$ r: p \rightarrow (\sim p \vee q) $$ has a truth value of false (F). We need to find the truth values of $$ p $$ and $$ q $$ that make $$ r $$ false.

Recall that an implication $$ a \rightarrow b $$ is false only when $$ a $$ is true and $$ b $$ is false. For $$ r $$ to be false, we must have:

• $$ p $$ is true (T), and
• $$ \sim p \vee q $$ is false (F).

Now, a disjunction (OR) like $$ \sim p \vee q $$ is false only when both components are false. Therefore:

• $$ \sim p $$ must be false (F), and
• $$ q $$ must be false (F).

If $$ \sim p $$ is false, then $$ p $$ must be true (T), because the negation of true is false and vice versa. This matches the first condition that $$ p $$ is true.

Thus, we have:

• $$ p = \text{T} $$
• $$ q = \text{F} $$

Let us verify these truth values by substituting them into $$ r $$.

If $$ p $$ is true (T), then $$ \sim p $$ is false (F).

Now, $$ \sim p \vee q = \text{F} \vee \text{F} = \text{F} $$ (since both are false).

Then, $$ r: p \rightarrow (\sim p \vee q) = \text{T} \rightarrow \text{F} $$.

We know that true implies false is false (T → F = F), which matches the given condition that $$ r $$ is false.

Therefore, the truth values of $$ p $$ and $$ q $$ are true (T) and false (F) respectively.

Looking at the options:

• A. F, F
• B. T, T
• C. T, F
• D. F, T

Option C matches T, F.

Hence, the correct answer is Option C.