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The question is asking which of the following logical expressions are tautologies. Let's analyze each one:

1. **\( p \lor 1 \)**:
   - This expression is a tautology. In logic, \( p \lor 1 \) means "p or true," and since one operand is always true, the entire expression will always evaluate to true, regardless of \( p \).

2. **\( (p \lor \neg p) \lor (\neg q \land q) \)**:
   - The first part, \( p \lor \neg p \), is the law of excluded middle, which is a tautology because it is always true (either \( p \) or \( \neg p \) must be true). The second part, \( \neg q \land q \), is a contradiction because \( q \) and \( \neg q \) cannot be true simultaneously. However, since it's connected by \( \lor \) (disjunction) with a tautology, the entire expression is still true. Therefore, this expression is also a tautology.

3. **\( p \lor 0 \)**:
   - This expression is **not** a tautology. \( p \lor 0 \) means "p or false," so the expression will be true only if \( p \) is true. It is not always true, so it is not a tautology.

4. **\( p \land \neg p \)**:
   - This expression is a **contradiction**. \( p \land \neg p \) means "p and not p," which can never be true because a statement and its negation cannot both be true simultaneously. Thus, it is not a tautology.

5. **\( p \land 1 \)**:
   - This expression is **not** a tautology. \( p \land 1 \) means "p and true," so the expression will evaluate to true only when \( p \) is true. It is not always true, so it is not a tautology.

6. **\( p \lor \neg p \)**:
   - This expression is a tautology. \( p \lor \neg p \) represents the law of excluded middle, meaning "p or not p," which is always true regardless of the truth value of \( p \).

### Summary of tautologies:
- \( p \lor 1 \)
- \( (p \lor \neg p) \lor (\neg q \land q) \)
- \( p \lor \neg p \)

Do you need any more details or have any questions?

Here are 5 related questions:
1. What is the difference between a tautology and a contradiction?
2. How do logical connectives like \( \land \) (and) and \( \lor \) (or) affect truth tables?
3. What is the significance of the law of excluded middle in logic?
4. Can you explain the process of identifying tautologies in complex expressions?
5. How do truth tables help in determining the truth value of a logical expression?

**Tip**: In propositional logic, a tautology is an expression that is true for every possible truth value of its variables.

Which expressions are tautologies? (Select all equivalent expressions.)

Understand how to identify tautologies in propositional logic. This problem involves evaluating logical expressions like p ∨ 1 and p ∨ ¬p and applying key theorems such as the law of excluded middle.

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