Key Concepts and Formulas

• The implication $A \to B$ is false only when $A$ is true and $B$ is false.
• The conjunction $A \wedge B$ is true only when both $A$ and $B$ are true.
• The disjunction $A \vee B$ is false only when both $A$ and $B$ are false.
• The negation of a true statement is false, and the negation of a false statement is true.

Step-by-Step Solution

Step 1: Analyze the given information.

We are given that $(p \wedge q) \to (\sim q \vee r)$ is false. We need to determine the truth values of $p$, $q$, and $r$.

Step 2: Apply the implication rule.

Since the implication $(p \wedge q) \to (\sim q \vee r)$ is false, it must be the case that the antecedent $(p \wedge q)$ is true and the consequent $(\sim q \vee r)$ is false. $(p \wedge q) = T$ $(\sim q \vee r) = F$

Step 3: Determine the truth values of $p$ and $q$.

Since $p \wedge q$ is true, both $p$ and $q$ must be true. $p = T$ $q = T$

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

Since $q$ is true, its negation $\sim q$ is false. $\sim q = F$

Step 5: Determine the truth value of $r$.

Since $\sim q \vee r$ is false, and we know that $\sim q$ is false, then $r$ must also be false. $r = F$

Step 6: Summarize the truth values of $p$, $q$, and $r$.

We have found that $p$ is true, $q$ is true, and $r$ is false. Therefore, the truth values of $p, q, r$ are T, T, F respectively.

Common Mistakes & Tips

• Remember the truth table for implication. $A \to B$ is only false when A is true and B is false.
• Be careful with negations. If $q$ is true, then $\sim q$ is false, and vice versa.
• It is useful to write out the truth tables for AND, OR, NOT, and IMPLICATION.

Summary

We are given that $(p \wedge q) \to (\sim q \vee r)$ is false. This means that $p \wedge q$ must be true and $\sim q \vee r$ must be false. Since $p \wedge q$ is true, both $p$ and $q$ must be true. Since $q$ is true, $\sim q$ is false. Since $\sim q \vee r$ is false and $\sim q$ is false, $r$ must also be false. Therefore, $p$ is true, $q$ is true, and $r$ is false.

Final Answer

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