https://file.aiquickdraw.com/mathbot/file/1729777569283az7k6lyz.jpg

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!

Determine the truth value of the compound statement given that P is a false statement, q is a true statement, and r is a true statement. [(P ∧ ¬q) ∨ ¬r] ∧ (P ∧ r)

Determine the truth value of a compound statement involving logical conjunction, disjunction, and negation. The problem includes evaluating the statement based on given truth values for P, q, and r. Suitable for students in Grades 10-12.

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