Which of the following statement is a tautology

Options

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

Answer

To determine which of the given statements is a tautology, we need to analyze each option and check if it is always true, regardless of the truth values of its components. A tautology is a statement that is true in every possible interpretation.

Let's denote the options as follows:
- Option A: \( \neg q \land p \land q \)
- Option B: \( \neg q \land p \lor p \land \neg p \)
- Option C: \( \neg q \land p \lor p \lor \neg p \)
- Option D: \( p \lor p \land q \land \neg p \land p \)

### Step-by-Step Solution:

**Step 1: Analyze Option A**
- The expression is \( \neg q \land p \land q \).
- According to the **Negation Law**, \( q \land \neg q \) is always false.
- Therefore, \( \neg q \land p \land q \) is also false for any values of \( p \) and \( q \).
- **Conclusion**: Option A is not a tautology.

**Step 2: Analyze Option B**
- The expression is \( \neg q \land p \lor p \land \neg p \).
- The term \( p \land \neg p \) is always false (again by **Negation Law**).
- Thus, the expression simplifies to \( \neg q \land p \).
- This expression can be true or false depending on the values of \( p \) and \( q \).
- **Conclusion**: Option B is not a tautology.

**Step 3: Analyze Option C**
- The expression is \( \neg q \land p \lor p \lor \neg p \).
- The term \( p \lor \neg p \) is always true (by the **Law of Excluded Middle**).
- Thus, the entire expression simplifies to true regardless of the values of \( p \) and \( q \).
- **Conclusion**: Option C is a tautology.

**Step 4: Analyze Option D**
- The expression is \( p \lor p \land q \land \neg p \land p \).
- The term \( p \land \neg p \) is always false.
- Thus, the expression simplifies to \( p \).
- This expression can be true or false depending on the value of \( p \).
- **Conclusion**: Option D is not a tautology.

### Final Answer:
The only option that is a tautology is **Option C**: \( \neg q \land p \lor p \lor \neg p \).