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 following statements is not a tautology, we will analyze each option by constructing truth tables for them. A tautology is a statement that is always true regardless of the truth values of its components.

### Step-by-Step Solution:

1. **Option 1: \( p \land q \implies p \lor q \)**

- Construct the truth table:
| p | q | \( p \land q \) | \( p \lor q \) | \( p \land q \implies p \lor q \) |
|-------|-------|------------------|-----------------|-----------------------------------|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | T |

- Conclusion: This statement is always true (tautology).

2. **Option 2: \( p \implies p \lor q \)**

- Construct the truth table:
| p | q | \( p \lor q \) | \( p \implies p \lor q \) |
|-------|-------|-----------------|-----------------------------|
| T | T | T | T |
| T | F | T | T |
| F | T | T | T |
| F | F | F | T |

- Conclusion: This statement is always true (tautology).

3. **Option 3: \( q \implies (p \implies q) \)**

- Construct the truth table:
| p | q | \( p \implies q \) | \( q \implies (p \implies q) \) |
|-------|-------|---------------------|----------------------------------|
| T | T | T | T |
| T | F | F | T |
| F | T | T | T |
| F | F | T | T |

- Conclusion: This statement is always true (tautology).

4. **Option 4: \( p \implies (p \land q) \)**

- Construct the truth table:
| p | q | \( p \land q \) | \( p \implies (p \land q) \) |
|-------|-------|------------------|-------------------------------|
| T | T | T | T |
| T | F | F | F |
| F | T | F | T |
| F | F | F | T |

- Conclusion: This statement is not always true (it is false when \( p \) is true and \( q \) is false).

### Final Answer:
The option that is not a tautology is **Option 4: \( p \implies (p \land q) \)**.