Key Concepts and Formulas

• The implication $p \to q$ is false only when $p$ is true and $q$ is false. Otherwise, it is true.
• The negation of a statement changes its truth value: if $p$ is true, then $\sim p$ is false, and vice versa.
• The disjunction $p \vee q$ is true if at least one of $p$ or $q$ is true. It is false only when both $p$ and $q$ are false.

Step-by-Step Solution

Step 1: Understand the given information.

We are given that the truth value of $p \to (\sim q \vee r)$ is false (F). This is our starting point. We need to find the truth values of $p$, $q$, and $r$ that make this statement false.

Step 2: Apply the condition for the falsity of an implication.

For the implication $p \to (\sim q \vee r)$ to be false, we must have $p$ being true (T) and $(\sim q \vee r)$ being false (F). This is because $p \to q$ is only false if $p$ is true and $q$ is false.

So, we have $p = T$ and $\sim q \vee r = F$.

Step 3: Analyze the disjunction.

For $\sim q \vee r$ to be false, both $\sim q$ and $r$ must be false. This is because the disjunction $a \vee b$ is only false if both $a$ and $b$ are false.

So, we have $\sim q = F$ and $r = F$.

Step 4: Determine the truth value of $q$.

Since $\sim q = F$, it means that $q$ must be true. This is because the negation of a false statement is true.

So, $q = T$.

Step 5: Summarize the truth values.

We have found that $p = T$, $q = T$, and $r = F$.

Common Mistakes & Tips

• Remember the truth table for implication ($p \to q$). It's only false when $p$ is true and $q$ is false.
• Be careful with negations. If $\sim q$ is false, then $q$ is true, and vice versa.
• When dealing with disjunctions ($p \vee q$), remember that it's only false when both $p$ and $q$ are false.

Summary

We are given that the statement $p \to (\sim q \vee r)$ is false. Using the truth table for implication, we deduce that $p$ must be true and $(\sim q \vee r)$ must be false. For the disjunction $(\sim q \vee r)$ to be false, both $\sim q$ and $r$ must be false. From $\sim q = F$, we get $q = T$. Therefore, the truth values of $p$, $q$, and $r$ are T, T, and F, respectively.

Final Answer

The final answer is \boxed{T, T, F}, which corresponds to option (C).