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Let's work through the logical expression \( p \lor (\neg p \lor q) \) by constructing a truth table and comparing it to the provided options.

### Steps for building the truth table:

1. List all possible truth values for \( p \) and \( q \).
2. Compute \( \neg p \) (negation of \( p \)).
3. Compute \( \neg p \lor q \) (disjunction of \( \neg p \) and \( q \)).
4. Finally, compute \( p \lor (\neg p \lor q) \).

#### Truth table construction:

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

- In the first row, \( p = T \) and \( q = T \), so \( \neg p = F \), and \( \neg p \lor q = T \), making \( p \lor (\neg p \lor q) = T \).
- In the second row, \( p = T \) and \( q = F \), so \( \neg p = F \), and \( \neg p \lor q = F \), making \( p \lor (\neg p \lor q) = T \).
- In the third row, \( p = F \) and \( q = T \), so \( \neg p = T \), and \( \neg p \lor q = T \), making \( p \lor (\neg p \lor q) = T \).
- In the fourth row, \( p = F \) and \( q = F \), so \( \neg p = T \), and \( \neg p \lor q = T \), making \( p \lor (\neg p \lor q) = T \).

#### Final column comparison:
The final column \( p \lor (\neg p \lor q) \) in the truth table is \( T, T, T, T \).

Now, let's compare the truth tables:

- Option **a**: Final column is \( T, T, T, T \) — this is the correct answer.
- Option **b**, **c**, and **d** have different results, so they are incorrect.

The correct answer is **a**.

Would you like more details or have any questions? Here are five related questions you can explore:

1. How do the logical operators \( \lor \) (OR) and \( \neg \) (NOT) function in truth tables?
2. Can you build a truth table for the expression \( (p \land q) \lor \neg p \)?
3. What happens if we change the logical expression to \( p \lor (\neg p \land q) \)?
4. How do we interpret complex logical expressions involving more than two variables?
5. What is the difference between logical equivalence and implication in truth tables?

**Tip**: When constructing truth tables, always handle the innermost parentheses and operations first to ensure correct results.

Construct a truth table and select the correct final column for the expression p ∨ (¬p ∨ q).

Construct a truth table for the logical expression p ∨ (¬p ∨ q). This example covers logical operators such as OR (∨) and NOT (¬), suitable for high school students learning about logic and truth tables.

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