https://file.aiquickdraw.com/mathbot/file/1729780648870kxpx1v9t.jpg

The problem in the image asks to determine the truth value of a compound statement given the following information:

- \( P \) is false (\( P = F \))
- \( Q \) is true (\( Q = T \))
- \( R \) is true (\( R = T \))

The compound statement is:
\[
\{[(P \cup \neg Q) \land R] \cup [(P \land \neg Q) \land \neg R] \} \cup [P \land (\neg Q \lor \neg R)]
\]

Let's break this down step-by-step using the truth values provided.

### Step 1: Break down the compound statement into parts

1. \( P \cup \neg Q \)
   - \( P = F \)
   - \( \neg Q = F \) (since \( Q = T \))
   - \( P \cup \neg Q = F \cup F = F \)

2. \( (P \cup \neg Q) \land R \)
   - From above, \( P \cup \neg Q = F \)
   - \( R = T \)
   - \( F \land T = F \)

3. \( P \land \neg Q \)
   - \( P = F \)
   - \( \neg Q = F \)
   - \( F \land F = F \)

4. \( (P \land \neg Q) \land \neg R \)
   - From above, \( P \land \neg Q = F \)
   - \( \neg R = F \) (since \( R = T \))
   - \( F \land F = F \)

5. \( \{[(P \cup \neg Q) \land R] \cup [(P \land \neg Q) \land \neg R]\} \)
   - Both parts are \( F \cup F = F \)

6. \( \neg Q \lor \neg R \)
   - \( \neg Q = F \)
   - \( \neg R = F \)
   - \( F \lor F = F \)

7. \( P \land (\neg Q \lor \neg R) \)
   - \( P = F \)
   - \( (\neg Q \lor \neg R) = F \)
   - \( F \land F = F \)

### Step 2: Evaluate the whole statement

Now the final expression is:
\[
F \cup F = F
\]

### Conclusion
The truth value of the compound statement is **false**.

Would you like more details or have any questions?

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Here are 5 relative questions based on the solution:

1. What are the truth tables for conjunction, disjunction, and negation?
2. How would the truth value change if \( P \) were true?
3. Can you simplify logical statements using De Morgan's laws?
4. What is the significance of parentheses in logical statements?
5. How would the compound statement be evaluated if \( Q \) were false?

**Tip:** When evaluating compound logical statements, always break them into smaller parts and solve each part step by step.

Determine the truth value of the compound statement [(P ∪ ¬Q) ∧ R] ∪ [(P ∧ ¬Q) ∧ ¬R] ∪ [P ∧ (¬Q ∨ ¬R)] given that P is false, Q is true, and R is true.

This page breaks down the evaluation of a compound logical statement using truth values: P is false, Q is true, and R is true. The key concepts include disjunction, conjunction, and negation. Suitable for college-level students studying propositional logic or discrete mathematics.

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