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The question asks you to use a truth table to determine which of the following statements is equivalent to \( p \lor (p \land q) \). The options are:

1. \( p \)
2. \( \neg p \lor \neg q \)
3. \( \neg q \)
4. \( p \lor q \)

### Step-by-Step Solution:

#### Expression: \( p \lor (p \land q) \)

Using the distributive property of logic:
\[
p \lor (p \land q) = p
\]
This is because \( p \land q \) will always be true when \( p \) is true, and \( p \lor (anything) \) simplifies to \( p \).

### Let's verify using a truth table:

| \( p \) | \( q \) | \( p \land q \) | \( p \lor (p \land q) \) | \( p \) |
|--------|--------|----------------|-------------------------|--------|
| T      | T      | T              | T                       | T      |
| T      | F      | F              | T                       | T      |
| F      | T      | F              | F                       | F      |
| F      | F      | F              | F                       | F      |

As seen, the column for \( p \lor (p \land q) \) is identical to the column for \( p \). Therefore, the statement \( p \lor (p \land q) \) is logically equivalent to \( p \).

#### Correct Answer: \( p \) (Option 1)

Would you like more details or have any questions?

### 5 Related Questions:
1. What is the distributive property in propositional logic?
2. How does \( p \land q \) affect the expression \( p \lor (p \land q) \)?
3. Can truth tables be used for more complex logical expressions?
4. What other logical equivalences can simplify expressions?
5. What is the relationship between conjunction (\( \land \)) and disjunction (\( \lor \)) in logic?

### Tip:
When simplifying logical expressions, always check for common logical identities like distributive, associative, and commutative properties to make your work easier.

Use a truth table to determine which of the following statements is equivalent to p ∨ (p ∧ q).

Learn how to determine the logical equivalence of p ∨ (p ∧ q) using a truth table. Explore the distributive property of logic and verify the expression's equivalence to p. This problem covers key concepts in propositional logic.

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