Which of the following is not a tautology?

Options

A. `(p^^q)rarr(pvvq)`
B. `prarr(pvvq)`
C. `qrarr(prarrq)`
D. `prarr(p^^q)`

Answer

To determine which of the given options is not a tautology, we will analyze each option by constructing their truth tables. A tautology is a statement that is true in every possible interpretation.

### Step-by-Step Solution:

1. **Understanding Tautology**: A tautology is a logical statement that is always true, regardless of the truth values of its components.

2. **Constructing Truth Tables**: We will create truth tables for each of the given options.

3. **Option A: \( P \land Q \)**:
- Truth Table:
| P | Q | \( P \land Q \) |
|-------|-------|------------------|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
- Result: Not a tautology (only true when both P and Q are true).

4. **Option B: \( P \Rightarrow (P \lor Q) \)**:
- Truth Table:
| P | Q | \( P \lor Q \) | \( P \Rightarrow (P \lor Q) \) |
|-------|-------|----------------|-------------------------------|
| True | True | True | True |
| True | False | True | True |
| False | True | True | True |
| False | False | False | True |
- Result: This is a tautology (always true).

5. **Option C: \( Q \Rightarrow (P \Rightarrow Q) \)**:
- Truth Table:
| P | Q | \( P \Rightarrow Q \) | \( Q \Rightarrow (P \Rightarrow Q) \) |
|-------|-------|-----------------------|--------------------------------------|
| True | True | True | True |
| True | False | False | True |
| False | True | True | True |
| False | False | True | True |
- Result: This is a tautology (always true).

6. **Option D: \( P \Rightarrow (P \land Q) \)**:
- Truth Table:
| P | Q | \( P \land Q \) | \( P \Rightarrow (P \land Q) \) |
|-------|-------|------------------|----------------------------------|
| True | True | True | True |
| True | False | False | False |
| False | True | False | True |
| False | False | False | True |
- Result: Not a tautology (false when P is true and Q is false).

7. **Conclusion**: The option that is not a tautology is **Option A** and **Option D**. However, since the question asks for one, we can conclude that **Option D** is the answer.

### Final Answer:
**Option D** is not a tautology.