Which of the following statements is not a tautology?

Options

A. `(p^^q ^^r)rArr (~pvv ~q vv r)`
B. `(p^^q ^^r) rArr ((~p^^~q)vvr)`
C. `(p^^~q ^^r) rArr (~pvv q vv r)`
D. `(p^^ q ^^ ~r) rArr r`

Answer

To determine which of the given statements is not a tautology, we need to analyze each statement by constructing truth tables and checking the truth values of each expression. A tautology is a statement that is always true, regardless of the truth values of its components.

### Step-by-Step Solution:

1. **Understand Tautology**: A tautology is a logical statement that is true in every possible interpretation. If any interpretation results in false, then it is not a tautology.

2. **Identify the Statements**: We have four logical expressions (let's denote them as A, B, C, D). We need to check each one to see if it is a tautology.

3. **Construct Truth Tables**: For each statement, we will create a truth table to evaluate the truth values.

4. **Evaluate Each Statement**:
- **Statement A**: \( p \land q \land r \implies \neg p \lor \neg q \lor r \)
- Create a truth table for \( p \), \( q \), and \( r \).
- Calculate \( p \land q \land r \).
- Calculate \( \neg p \), \( \neg q \), and \( \neg p \lor \neg q \lor r \).
- Check the implication \( p \land q \land r \implies \neg p \lor \neg q \lor r \).
- Result: All values are true, so this is a tautology.

- **Statement B**: \( p \land q \land r \land \neg q \land \neg r \)
- Create a truth table.
- Calculate \( p \land q \land r \) and \( \neg q \land \neg r \).
- Check the conjunction.
- Result: This statement has false values, so it is not a tautology.

- **Statement C**: \( p \land \neg q \land r \implies \neg p \lor q \lor r \)
- Create a truth table.
- Calculate \( p \land \neg q \land r \).
- Calculate \( \neg p \lor q \lor r \).
- Check the implication.
- Result: All values are true, so this is a tautology.

- **Statement D**: \( p \land q \land \neg r \implies r \)
- Create a truth table.
- Calculate \( p \land q \land \neg r \).
- Check the implication with \( r \).
- Result: This statement has false values, so it is not a tautology.

5. **Conclusion**: After evaluating all statements, we find that Statements B and D are not tautologies. However, the question asks for one statement that is not a tautology. Therefore, we can conclude that **Statement B** is one of the statements that is not a tautology.

### Final Answer:
The statement that is not a tautology is **Statement B**.