Which of the following is a contradiction

Options

A. `(p ^^q) ^^~(p vvq)`
B. `p vv(~p ^^q)`
C. `(p implies q) impliesp`
D. None of these

Answer

To determine which of the given statements is a contradiction, we will analyze each option using truth tables. A contradiction is a statement that is always false, regardless of the truth values of its components.

### Step-by-Step Solution:

1. **Identify the Statements**: We have three options to evaluate:
- Option 1: \( P \land Q \)
- Option 2: \( \neg P \land (P \lor Q) \)
- Option 3: \( P \implies Q \land \neg P \)

2. **Create a Truth Table**: We will create a truth table for each option to evaluate their truth values.

**Truth Table Structure**:
- Columns for \( P \), \( Q \), \( \neg P \), \( P \land Q \), \( P \lor Q \), \( P \implies Q \), and the final expressions for each option.

| \( P \) | \( Q \) | \( \neg P \) | \( P \land Q \) | \( P \lor Q \) | \( P \implies Q \) | Option 1 | Option 2 | Option 3 |
|---------|---------|---------------|------------------|-----------------|---------------------|----------|----------|----------|
| T | T | F | T | T | T | T | F | T |
| T | F | F | F | T | F | F | F | F |
| F | T | T | F | T | T | F | T | T |
| F | F | T | F | F | T | F | F | T |

3. **Evaluate Each Option**:
- **Option 1**: \( P \land Q \)
- The truth values are: T, F, F, F.
- This option is **not a contradiction** because it is true when both \( P \) and \( Q \) are true.

- **Option 2**: \( \neg P \land (P \lor Q) \)
- The truth values are: F, F, T, F.
- This option is **not a contradiction** because it is true when \( P \) is false and \( Q \) is true.

- **Option 3**: \( P \implies Q \land \neg P \)
- The truth values are: T, F, T, T.
- This option is **not a contradiction** because it is true in some cases.

4. **Conclusion**: After evaluating all options, we find that none of the options are contradictions. However, if we were to find a contradiction, we would look for a statement that is always false.

### Final Answer:
None of the given options is a contradiction.