Which of the following is false ?

Options

A. `p vv ~ p` is a tautology
B. ` ~ ( ~ p ) harr p` is a tautology
C. ` ( p ^^ ( p to q)) to p ` is a contradiction
D. `p ^^ ~ p ` is a contradiction

Answer

To determine which of the given statements is false, we will analyze each option step by step.

### Step 1: Analyze Option 1 - \( P \lor \neg P \)

1. **Truth Table Construction**:
- Let \( P \) be true (T) and false (F).
- \( \neg P \) will be false (F) when \( P \) is true and true (T) when \( P \) is false.
- The expression \( P \lor \neg P \) will be true in both cases.

| P | ¬P | P ∨ ¬P |
|-------|-------|--------|
| T | F | T |
| F | T | T |

2. **Conclusion**: Since \( P \lor \neg P \) is always true, it is a tautology.

### Step 2: Analyze Option 2 - \( \neg(\neg P) \leftrightarrow P \)

1. **Simplification**:
- \( \neg(\neg P) \) simplifies to \( P \).
- Thus, the expression becomes \( P \leftrightarrow P \), which is always true.

2. **Conclusion**: This statement is also a tautology.

### Step 3: Analyze Option 3 - \( P \land (P \implies Q) \implies P \)

1. **Understanding the Implication**:
- The implication \( P \implies Q \) can be rewritten as \( \neg P \lor Q \).
- Therefore, \( P \land (P \implies Q) \) becomes \( P \land (\neg P \lor Q) \).

2. **Distributing**:
- Using the distributive property:
\[ P \land (\neg P \lor Q) = (P \land \neg P) \lor (P \land Q) \]
- \( P \land \neg P \) is always false.

3. **Final Expression**:
- Thus, we have \( \text{false} \lor (P \land Q) \), which simplifies to \( P \land Q \).

4. **Implication**:
- The expression \( (P \land Q) \implies P \) is not always true; it depends on the values of \( P \) and \( Q \).

5. **Conclusion**: This statement is not a tautology and can be false.

### Step 4: Analyze Option 4 - \( P \land \neg P \)

1. **Truth Table Construction**:
- \( P \land \neg P \) is always false because both cannot be true at the same time.

| P | ¬P | P ∧ ¬P |
|-------|-------|--------|
| T | F | F |
| F | T | F |

2. **Conclusion**: This statement is a contradiction.

### Final Conclusion

- **Option 1**: True (Tautology)
- **Option 2**: True (Tautology)
- **Option 3**: False (Not a tautology)
- **Option 4**: True (Contradiction)

Thus, the false statement is **Option 3**.

---