Key Concepts and Formulas

• Implication ($p \to q$): The implication $p \to q$ is false only when $p$ is true and $q$ is false. In all other cases, it is true.
• Conjunction ($p \wedge q$): The conjunction $p \wedge q$ is true only when both $p$ and $q$ are true. Otherwise, it is false.
• Negation ($\sim q$): The negation $\sim q$ is true when $q$ is false, and false when $q$ is true.

Step-by-Step Solution

Step 1: Analyze the given statement We are given that $p \to (p \wedge \sim q)$ is false.

Step 2: Apply the rule for implication For the implication $p \to (p \wedge \sim q)$ to be false, we must have $p$ as true and $(p \wedge \sim q)$ as false. This is because $p \to q$ is only false when $p$ is true and $q$ is false. Therefore, $p = T$ and $(p \wedge \sim q) = F$.

Step 3: Apply the rule for conjunction Since $p \wedge \sim q$ is false, it means that it is not the case that both $p$ and $\sim q$ are true. We already know that $p$ is true. Therefore, $\sim q$ must be false.

Step 4: Apply the rule for negation If $\sim q$ is false, then $q$ must be true.

Step 5: Summarize the truth values We have $p = T$ and $q = T$.

Common Mistakes & Tips

• Remember the truth table for implication. The only case where $p \to q$ is false is when $p$ is true and $q$ is false.
• Be careful with negation. If $\sim q$ is false, then $q$ is true, and vice versa.
• When dealing with complex logical statements, break them down into smaller parts and analyze each part separately.

Summary

We are given that $p \to (p \wedge \sim q)$ is false. This means that $p$ must be true and $p \wedge \sim q$ must be false. Since $p$ is true, for $p \wedge \sim q$ to be false, $\sim q$ must be false. Therefore, $q$ must be true. Thus, $p = T$ and $q = T$.

Final Answer The final answer is \boxed{T, T}, which corresponds to option (A).