Which of the following is a tautology ?

Options

A. ` p ^^ q`
B. ` p vv q`
C. ` p vv ~ p`
D. ` p ^^ ~ p `

Answer

To determine which of the given options is a tautology, we will analyze each option using truth tables. A tautology is a statement that is always true regardless of the truth values of its components.

### Step 1: Define the Options
We have four options:
1. \( p \land q \)
2. \( p \lor q \)
3. \( p \lor \neg p \)
4. \( p \land \neg p \)

### Step 2: Create a Truth Table
We will create a truth table for each option, considering all possible truth values for \( p \) and \( q \).

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

### Step 3: Analyze Each Option
1. **Option A: \( p \land q \)**
- Truth values: T, F, F, F
- Not a tautology (not always true).

2. **Option B: \( p \lor q \)**
- Truth values: T, T, T, F
- Not a tautology (not always true).

3. **Option C: \( p \lor \neg p \)**
- Truth values: T, T, T, T
- This is a tautology (always true).

4. **Option D: \( p \land \neg p \)**
- Truth values: F, F, F, F
- Not a tautology (always false).

### Conclusion
The only option that is a tautology is **Option C: \( p \lor \neg p \)**.

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