We are given a compound statement:

$[(P \land \neg Q) \lor \neg R] \land (P \land R)$

with the following truth values:

• $P$ is false,
• $Q$ is true,
• $R$ is true.

Step-by-Step Evaluation:

1. Evaluate $P \land \neg Q$:

   • $P$ is false.
   • $Q$ is true, so $\neg Q$ is false.
   • Therefore, $P \land \neg Q = \text{false} \land \text{false} = \text{false}$.

2. Evaluate $\neg R$:

   • $R$ is true, so $\neg R$ is false.

3. Combine $(P \land \neg Q) \lor \neg R$:

   • We have $\text{false} \lor \text{false}$, which equals false.

4. Evaluate $P \land R$:

   • $P$ is false.
   • $R$ is true.
   • So, $P \land R = \text{false} \land \text{true} = \text{false}$.

5. Final Compound Statement Evaluation:

   • Now, we have $[(P \land \neg Q) \lor \neg R] \land (P \land R)$.
   • Substitute the evaluated values: $\text{false} \land \text{false}$ equals false.

Conclusion:

The compound statement evaluates to false.

Would you like further details or explanations on any specific step? Here are some related questions:

1. How do truth tables help in evaluating complex compound statements?
2. What is the role of conjunction ($\land$) and disjunction ($\lor$) in logic?
3. What happens if $P$, $Q$, and $R$ all have the same truth value?
4. What is the significance of negation ($\neg$) in changing a statement’s truth value?
5. Can we simplify the expression to directly find the truth value?

Tip: When dealing with compound statements, always break them down into smaller parts and evaluate each part step-by-step. This minimizes errors and helps in better understanding!