Which one of the following Boolean expressions is a tautology?

Options

A. `(p vee q) wedge (pv ~q)`
B. `(p wedge q) vee (p wedge ~q)`
C. `(p vee q) wedge (~pv ~q)`
D. `(p vee q) vee (pv ~q)`

Explanation

To determine which of the given Boolean expressions is a tautology, we will evaluate each option systematically by constructing truth tables. A tautology is a statement that is always true regardless of the truth values of its variables.

### Step-by-Step Solution:

1. **Identify the Options**:
We need to evaluate the following Boolean expressions:
- Option 1: \( P \lor Q \lor (P \lor \neg P) \lor \neg Q \)
- Option 2: \( (P \land Q) \lor (P \land \neg Q) \)
- Option 3: \( (P \lor Q) \land (\neg P \lor \neg Q) \)
- Option 4: \( P \lor Q \lor (P \lor \neg Q) \)

2. **Construct Truth Tables**:
We will create truth tables for each option.

**Option 1**: \( P \lor Q \lor (P \lor \neg P) \lor \neg Q \)
- \( P \) | \( Q \) | \( \neg P \) | \( \neg Q \) | \( P \lor \neg P \) | \( P \lor Q \) | Final Result
- T | T | F | F | T | T | T
- T | F | F | T | T | T | T
- F | T | T | F | T | T | T
- F | F | T | T | T | F | T

**Conclusion**: Option 1 is a tautology (always true).

**Option 2**: \( (P \land Q) \lor (P \land \neg Q) \)
- \( P \) | \( Q \) | \( \neg Q \) | \( P \land Q \) | \( P \land \neg Q \) | Final Result
- T | T | F | T | F | T
- T | F | T | F | T | T
- F | T | F | F | F | F
- F | F | T | F | F | F

**Conclusion**: Option 2 is not a tautology (not always true).

**Option 3**: \( (P \lor Q) \land (\neg P \lor \neg Q) \)
- \( P \) | \( Q \) | \( \neg P \) | \( \neg Q \) | \( P \lor Q \) | \( \neg P \lor \neg Q \) | Final Result
- T | T | F | F | T | F | F
- T | F | F | T | T | T | T
- F | T | T | F | T | T | T
- F | F | T | T | F | T | F

**Conclusion**: Option 3 is not a tautology (not always true).

**Option 4**: \( P \lor Q \lor (P \lor \neg Q) \)
- \( P \) | \( Q \) | \( \neg Q \) | \( P \lor \neg Q \) | Final Result
- T | T | F | T | T
- T | F | T | T | T
- F | T | F | F | T
- F | F | T | T | T

**Conclusion**: Option 4 is a tautology (always true).

3. **Final Answer**:
The Boolean expressions that are tautologies are **Option 1** and **Option 4**.