Which of these is false :

Options

A. `~| p ^^ p`
B. `p vv q`
C. `p ^^ q`
D. `p implies q`

Answer

To determine which of the given options is false, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Options**:
We have four options to evaluate:
- Option A: Negation P, conjunction P (¬P ∧ P)
- Option B: P disjunction Q (P ∨ Q)
- Option C: P conjunction Q (P ∧ Q)
- Option D: P implies Q (P → Q)

2. **Evaluating Option A (¬P ∧ P)**:
- The expression ¬P ∧ P represents the conjunction of P and its negation.
- In logical terms, this means both P is true and P is false at the same time, which is impossible.
- Therefore, this expression is always false (it has a truth value of 0).

3. **Evaluating Option B (P ∨ Q)**:
- The expression P ∨ Q is a disjunction, meaning it is true if at least one of P or Q is true.
- Depending on the truth values of P and Q, this expression can be either true or false. Hence, it is not always false.

4. **Evaluating Option C (P ∧ Q)**:
- The expression P ∧ Q is a conjunction, meaning it is true only if both P and Q are true.
- Similar to option B, this expression can also be either true or false depending on the values of P and Q. Thus, it is not always false.

5. **Evaluating Option D (P → Q)**:
- The expression P → Q is an implication, which is false only when P is true and Q is false.
- In all other cases, it is true. Therefore, this expression can also be either true or false, and is not always false.

6. **Conclusion**:
- After evaluating all options, we find that Option A (¬P ∧ P) is the only expression that is always false.

### Final Answer:
The false option is **Option A: Negation P, conjunction P (¬P ∧ P)**.