Which of the following statements is a tautology?

Options

A. ` ( ~ p vv q) ~ ( p vv ~ q)`
B. ` ( ~ p vv ~ q) to p vv q`
C. ` ( p vv ~ q) ^^ ( p vv q)`
D. ` ( ~ p vv ~ q) vv ( p vv q)`

Explanation

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

### Step-by-Step Solution:

1. **Identify the Statements**:
The options provided are:
- Option 1: ¬p ∨ q → p ∨ ¬q
- Option 2: ¬p ∨ ¬q → p ∨ q
- Option 3: p ∨ ¬q ∧ p ∨ q
- Option 4: ¬(p ∨ ¬q) ∨ (p ∨ q)

2. **Evaluate Option 1: ¬p ∨ q → p ∨ ¬q**
- Construct a truth table for the components:
- p | q | ¬p | ¬q | ¬p ∨ q | p ∨ ¬q | ¬p ∨ q → p ∨ ¬q
- T | T | F | F | T | T | T
- T | F | F | T | F | T | T
- F | T | T | F | T | F | F
- F | F | T | T | T | T | T
- The result is not always true (False for p = F, q = T). Thus, it is **not a tautology**.

3. **Evaluate Option 2: ¬p ∨ ¬q → p ∨ q**
- Construct a truth table:
- p | q | ¬p | ¬q | ¬p ∨ ¬q | p ∨ q | ¬p ∨ ¬q → p ∨ q
- T | T | F | F | F | T | T
- T | F | F | T | T | T | T
- F | T | T | F | T | T | T
- F | F | T | T | T | F | F
- The result is not always true (False for p = F, q = F). Thus, it is **not a tautology**.

4. **Evaluate Option 3: p ∨ ¬q ∧ p ∨ q**
- Construct a truth table:
- p | q | ¬q | ¬q ∧ p | p ∨ ¬q | p ∨ q | p ∨ ¬q ∧ p ∨ q
- T | T | F | F | T | T | T
- T | F | T | T | T | T | T
- F | T | F | F | F | T | T
- F | F | T | F | F | F | F
- The result is not always true (False for p = F, q = F). Thus, it is **not a tautology**.

5. **Evaluate Option 4: ¬(p ∨ ¬q) ∨ (p ∨ q)**
- Construct a truth table:
- p | q | ¬q | p ∨ ¬q | p ∨ q | ¬(p ∨ ¬q) | ¬(p ∨ ¬q) ∨ (p ∨ q)
- T | T | F | T | T | F | T
- T | F | T | T | T | F | T
- F | T | F | F | T | T | T
- F | F | T | T | F | F | T
- The result is always true. Thus, it **is a tautology**.

### Conclusion:
The statement that is a tautology is **Option 4: ¬(p ∨ ¬q) ∨ (p ∨ q)**.