`p ^^ ~| p` is :

Options

A. False
B. True
C. p
D. q

Answer

To solve the expression \( p \land \neg p \) (which reads as "p and not p"), we can follow these steps:

### Step 1: Understand the symbols
The expression \( p \land \neg p \) consists of:
- \( p \): a proposition that can either be true (T) or false (F).
- \( \neg p \): the negation of \( p \), which is true when \( p \) is false and false when \( p \) is true.
- \( \land \): the logical conjunction (AND) operator, which is true only when both operands are true.

**Hint for Step 1:** Identify the components of the expression and their meanings.

### Step 2: Analyze the conjunction
The conjunction \( p \land \neg p \) can be evaluated based on the truth values of \( p \):
- If \( p \) is true (T), then \( \neg p \) is false (F). Therefore, \( p \land \neg p = T \land F = F \).
- If \( p \) is false (F), then \( \neg p \) is true (T). Therefore, \( p \land \neg p = F \land T = F \).

**Hint for Step 2:** Evaluate the expression for both possible truth values of \( p \).

### Step 3: Conclusion
In both cases, whether \( p \) is true or false, the expression \( p \land \neg p \) evaluates to false (F). Hence, the overall value of \( p \land \neg p \) is false.

**Hint for Step 3:** Summarize the results of your evaluation to reach a conclusion.

### Final Answer
The expression \( p \land \neg p \) is always false.