A tautology is a logical statement that is always true, regardless of the truth values of its components. We are asked to show that the statement \( p \lor \neg p \) is a tautology.

### Statement to Prove:
\[ p \lor \neg p \]

This expression reads as "p OR not p." To prove this, we will analyze it using a **truth table** for all possible truth values of \( p \).

### Truth Table:

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

- When \( p \) is true (T), \( \neg p \) is false (F), and \( p \lor \neg p \) is true.
- When \( p \) is false (F), \( \neg p \) is true (T), and \( p \lor \neg p \) is still true.

Since the statement \( p \lor \neg p \) is **true in all cases**, it is a **tautology**.

### Conclusion:
The expression \( p \lor \neg p \) is always true, making it a tautology.

Would you like more details on tautologies, or do you have any further questions?

Here are 5 related questions:
1. What are other examples of tautologies in logic?
2. Can a tautology be expressed using conjunctions instead of disjunctions?
3. What is the difference between a tautology and a contradiction?
4. How can tautologies be useful in mathematical proofs?
5. How does \( p \lor \neg p \) relate to the Law of the Excluded Middle?

**Tip:** Tautologies are often used in logic and proofs to establish statements that hold universally, regardless of individual truth values.

Discover how to prove that p ∨ ¬p is a tautology using a truth table. This problem explains key concepts in propositional logic, such as logical disjunction and the Law of the Excluded Middle. Suitable for high school students in Grades 10-12.

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