Construct the truth table of the compound proposition ( ) ( ). p q p q ∨ ¬ → ∧ 

The compound proposition you provided is a bit unclear due to formatting, but I understand it as:

\[
(p \lor \neg q) \rightarrow (p \land q)
\]

We need to construct the truth table for this expression. Let's break it down step by step.

### Components:
1. **\( p \)**: Proposition \( p \).
2. **\( q \)**: Proposition \( q \).
3. **\( \neg q \)**: The negation of \( q \).
4. **\( p \lor \neg q \)**: \( p \) OR \( \neg q \).
5. **\( p \land q \)**: \( p \) AND \( q \).
6. **\( (p \lor \neg q) \rightarrow (p \land q) \)**: The implication from \( p \lor \neg q \) to \( p \land q \).

### Truth Table Construction:
Let's construct the truth table by considering all possible truth values for \( p \) and \( q \).

| \( p \) | \( q \) | \( \neg q \) | \( p \lor \neg q \) | \( p \land q \) | \( (p \lor \neg q) \rightarrow (p \land q) \) |
|:---:|:---:|:---:|:---:|:---:|:---:|
| T | T | F | T | T | T |
| T | F | T | T | F | F |
| F | T | F | F | F | T |
| F | F | T | T | F | F |

### Explanation:

1. **First Row**:  
   - \( p = T \), \( q = T \)
   - \( \neg q = F \)
   - \( p \lor \neg q = T \lor F = T \)
   - \( p \land q = T \land T = T \)
   - \( (p \lor \neg q) \rightarrow (p \land q) = T \rightarrow T = T \)

2. **Second Row**:  
   - \( p = T \), \( q = F \)
   - \( \neg q = T \)
   - \( p \lor \neg q = T \lor T = T \)
   - \( p \land q = T \land F = F \)
   - \( (p \lor \neg q) \rightarrow (p \land q) = T \rightarrow F = F \)

3. **Third Row**:  
   - \( p = F \), \( q = T \)
   - \( \neg q = F \)
   - \( p \lor \neg q = F \lor F = F \)
   - \( p \land q = F \land T = F \)
   - \( (p \lor \neg q) \rightarrow (p \land q) = F \rightarrow F = T \)

4. **Fourth Row**:  
   - \( p = F \), \( q = F \)
   - \( \neg q = T \)
   - \( p \lor \neg q = F \lor T = T \)
   - \( p \land q = F \land F = F \)
   - \( (p \lor \neg q) \rightarrow (p \land q) = T \rightarrow F = F \)

### Final Truth Table:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
p & q & \neg q & p \lor \neg q & p \land q & (p \lor \neg q) \rightarrow (p \land q) \\
\hline
T & T & F & T & T & T \\
T & F & T & T & F & F \\
F & T & F & F & F & T \\
F & F & T & T & F & F \\
\hline
\end{array}
\]

Would you like any more details or have any questions?

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Here are 5 questions related to this topic:

1. What is the difference between conjunction (\(\land\)) and disjunction (\(\lor\)) in logic?
2. How does the truth value of an implication (\(\rightarrow\)) change based on its premises?
3. Can you create a truth table for \( p \rightarrow (q \lor r) \)?
4. What is the role of negation (\(\neg\)) in logical statements?
5. How does the truth table change if we reverse the implication, i.e., \( (p \land q) \rightarrow (p \lor \neg q) \)?

**Tip:** In logic, an implication (\(\rightarrow\)) is false only when the first part (antecedent) is true, and the second part (consequent) is false.

Learn how to construct the truth table for the compound proposition (p ∨ ¬q) → (p ∧ q). This problem involves propositional logic, truth table construction, and logical implications. Suitable for advanced high school students and anyone interested in formal logic.

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