Use a truth table to determine whether the given statement is a tautology.
p ∨ ~p

To determine whether the statement \( p \lor \neg p \) (read as "p or not p") is a tautology, we can construct a truth table.

A tautology is a statement that is always true, regardless of the truth values of its components. Let's analyze the truth values of \( p \) and \( \neg p \), and then evaluate \( p \lor \neg p \).

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

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

Would you like a further breakdown or details on how this works?

Here are 5 related questions to expand on this:

1. How do you construct a truth table for more complex logical statements?
2. What are examples of statements that are not tautologies?
3. How do we distinguish between a tautology, a contradiction, and a contingency?
4. What are De Morgan's laws and how do they relate to logical operations?
5. How can you simplify a compound logical statement using truth tables?

**Tip:** A tautology always produces a truth value of "true" regardless of the input variables, making it very useful in logical proofs.

Use a truth table to determine whether the given statement is a tautology. p ∨ ~p

Explore how to use a truth table to determine whether the logical statement p ∨ ~p (p or not p) is a tautology. This detailed guide covers key concepts in propositional logic, including the Law of Excluded Middle, suitable for Grades 9-12.

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