Which of the following statements is a tautology ?

Options

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

Explanation

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

Let's denote the options as follows:
1. \( \neg Q \land P \)
2. \( \neg Q \land P \land \neg P \)
3. \( \neg P \lor P \lor \neg Q \lor P \)
4. \( P \land Q \)

Now, we will evaluate each option step by step.

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

**Truth Table:**
| P | Q | \( \neg Q \) | \( \neg Q \land P \) |
|---|---|--------------|-----------------------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |

**Conclusion:** This statement is not always true (it is false when \( P \) is false or \( Q \) is true). Therefore, it is not a tautology.

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

**Truth Table:**
| P | Q | \( \neg Q \) | \( \neg P \) | \( \neg Q \land P \land \neg P \) |
|---|---|--------------|---------------|-------------------------------------|
| T | T | F | F | F |
| T | F | T | F | F |
| F | T | F | T | F |
| F | F | T | T | F |

**Conclusion:** This statement is also not always true (it is false in all cases). Therefore, it is not a tautology.

### Step 3: Analyze Option 3 - \( \neg P \lor P \lor \neg Q \lor P \)

**Truth Table:**
| P | Q | \( \neg P \) | \( \neg Q \) | \( \neg P \lor P \) | \( \neg P \lor P \lor \neg Q \lor P \) |
|---|---|--------------|--------------|-----------------------|-----------------------------------------|
| T | T | F | F | T | T |
| T | F | F | T | T | T |
| F | T | T | F | T | T |
| F | F | T | T | T | T |

**Conclusion:** This statement is always true regardless of the truth values of \( P \) and \( Q \). Therefore, it is a tautology.

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

**Truth Table:**
| P | Q | \( P \land Q \) |
|---|---|------------------|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |

**Conclusion:** This statement is not always true (it is false when either \( P \) or \( Q \) is false). Therefore, it is not a tautology.

### Final Conclusion:
The only option that is a tautology is **Option 3: \( \neg P \lor P \lor \neg Q \lor P \)**.

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