Which of the following propositions is a tautology ?

Options

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

Answer

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

### Step-by-Step Solution:

1. **Identify the Propositions**: Let's denote the propositions as follows:
- Proposition 1: \( m_1 \)
- Proposition 2: \( m_2 \)
- Proposition 3: \( m_3 \)
- Proposition 4: \( m_4 \)

2. **Determine the Variables**: The variables involved are \( p \) and \( q \). Since there are two variables, we can have \( 2^2 = 4 \) possible combinations of truth values.

3. **Create the Truth Table**: We will create a truth table for \( p \) and \( q \):
- \( p \): True (T), True (T), False (F), False (F)
- \( q \): True (T), False (F), True (T), False (F)

The truth table will look like this:

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

4. **Evaluate Each Proposition**:
- **For \( m_1 \)**: \( \neg p \lor \neg q \)
- Evaluate \( \neg p \lor \neg q \):
- Row 1: F ∨ F = F
- Row 2: F ∨ T = T
- Row 3: T ∨ F = T
- Row 4: T ∨ T = T
- Result: F, T, T, T (Not a tautology)

- **For \( m_2 \)**: \( p \lor \neg q \)
- Evaluate \( p \lor \neg q \):
- Row 1: T ∨ F = T
- Row 2: T ∨ T = T
- Row 3: F ∨ F = F
- Row 4: F ∨ T = T
- Result: T, T, F, T (Not a tautology)

- **For \( m_3 \)**: \( \neg p \land (\neg p \lor \neg q) \)
- Evaluate \( \neg p \land (\neg p \lor \neg q) \):
- Row 1: F ∧ F = F
- Row 2: F ∧ T = F
- Row 3: T ∧ T = T
- Row 4: T ∧ T = T
- Result: F, F, T, T (Not a tautology)

- **For \( m_4 \)**: \( \neg q \land (p \lor \neg q) \)
- Evaluate \( \neg q \land (p \lor \neg q) \):
- Row 1: F ∧ T = F
- Row 2: T ∧ T = T
- Row 3: F ∧ F = F
- Row 4: T ∧ T = T
- Result: F, T, F, T (Not a tautology)

5. **Conclusion**: After evaluating all propositions, we find that none of the propositions are tautologies. However, if we assume that \( m_1 \) is the only one that consistently returns true in most cases, we can say it is the closest to being a tautology, but it is not a tautology since it has a false outcome.

### Final Answer:
None of the propositions are tautologies.