which of the following is a contradiction ?

Options

A. ` p vv q`
B. ` P ^^ q`
C. ` p vv ~ p`
D. ` p ^^ ~ p `

Answer

To determine which of the given options 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 Variables**: Let's denote two propositions as \( p \) and \( q \).

2. **Construct the Truth Table**: We will create a truth table for \( p \) and \( q \) to evaluate the logical expressions. The truth values for \( p \) and \( q \) can be represented as follows:

| \( p \) | \( q \) |
|---------|---------|
| T | T |
| T | F |
| F | T |
| F | F |

3. **Evaluate Each Option**:
- **Option A**: \( p \cup q \) (Logical OR)
- This expression is true if at least one of \( p \) or \( q \) is true.
- Truth values:
- T (T or T)
- T (T or F)
- T (F or T)
- F (F or F)
- Conclusion: Not a contradiction (true in some cases).

- **Option B**: \( p \cap q \) (Logical AND)
- This expression is true only if both \( p \) and \( q \) are true.
- Truth values:
- T (T and T)
- F (T and F)
- F (F and T)
- F (F and F)
- Conclusion: Not a contradiction (true in one case).

- **Option C**: \( p \cup \neg p \) (Logical OR with negation)
- This expression is always true because either \( p \) is true or \( p \) is false.
- Truth values:
- T (T or F)
- T (T or F)
- T (F or T)
- T (F or T)
- Conclusion: Not a contradiction (always true).

- **Option D**: \( p \cap \neg p \) (Logical AND with negation)
- This expression is true only if both \( p \) is true and \( p \) is false, which is impossible.
- Truth values:
- F (T and F)
- F (T and F)
- F (F and T)
- F (F and T)
- Conclusion: This is a contradiction (always false).

4. **Final Answer**: The option that is a contradiction is **Option D**.