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Let's construct the truth table for the given logical statement:

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

To solve this, we will evaluate step by step. The columns you need to fill are:

1. \(\neg p\) (the negation of \(p\))
2. \(q \lor \neg p\) (the disjunction of \(q\) and \(\neg p\))
3. \(p \land (q \lor \neg p)\) (the conjunction of \(p\) with the previous result)
4. \([p \land (q \lor \neg p)] \lor r\) (the final disjunction with \(r\))

Let's break this down row by row, based on all possible truth values for \(p\), \(q\), and \(r\) (True = T, False = F):

| \(p\) | \(q\) | \(r\) | \(\neg p\) | \(q \lor \neg p\) | \(p \land (q \lor \neg p)\) | \([p \land (q \lor \neg p)] \lor r\) |
|------|-------|------|-----------|-----------------|-------------------------|----------------------------------|
|  T   |   T   |  T   |     F     |        T        |            T            |               T                  |
|  T   |   T   |  F   |     F     |        T        |            T            |               T                  |
|  T   |   F   |  T   |     F     |        F        |            F            |               T                  |
|  T   |   F   |  F   |     F     |        F        |            F            |               F                  |
|  F   |   T   |  T   |     T     |        T        |            F            |               T                  |
|  F   |   T   |  F   |     T     |        T        |            F            |               F                  |
|  F   |   F   |  T   |     T     |        T        |            F            |               T                  |
|  F   |   F   |  F   |     T     |        T        |            F            |               F                  |

### Detailed Explanation:
1. **\(\neg p\)** is just the negation of \(p\).
2. **\(q \lor \neg p\)** combines the value of \(q\) and the negation of \(p\) using logical "or."
3. **\(p \land (q \lor \neg p)\)** takes the value of \(p\) and the result of \(q \lor \neg p\) using logical "and."
4. **\([p \land (q \lor \neg p)] \lor r\)** combines the previous result with \(r\) using logical "or."

Would you like a more detailed explanation or have any questions about this?

Here are 5 follow-up questions related to this topic:
1. How do you determine the negation of a variable in a truth table?
2. What is the difference between conjunction (\(\land\)) and disjunction (\(\lor\))?
3. How do you know when a complex statement is true or false?
4. Can you find a scenario where a complex logical statement simplifies to a single truth value?
5. How would the truth table change if one of the operators was replaced with a conditional (→)?

**Tip**: Breaking complex logical statements into smaller components can make evaluating them easier, especially when building truth tables.

Construct a truth table for the given logical statement [p ∧ (q ∨ ¬p)] ∨ r.

Learn how to construct a truth table for the logical expression [p ∧ (q ∨ ¬p)] ∨ r. This problem involves key logical concepts such as negation, conjunction, and disjunction, with a step-by-step solution. Suitable for high school students in Grades 10-12.

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