`p wedgeq` is true when `:`

Options

A. p and q are both true
B. p and q are both false
C. p is true and q is false
D. p is false and q is true

Answer

To determine when the expression \( p \wedge q \) (which represents the logical "AND" operation) is true, we need to analyze the truth values of the propositions \( p \) and \( q \).

### Step-by-Step Solution:

1. **Understanding the Logical AND Operation**:
The logical AND operation, denoted as \( p \wedge q \), is true only when both \( p \) and \( q \) are true.

2. **Creating a Truth Table**:
We will create a truth table to evaluate all possible combinations of truth values for \( p \) and \( q \).

| \( p \) | \( q \) | \( p \wedge q \) |
|---------|---------|-------------------|
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |

3. **Evaluating Each Case**:
- When both \( p \) and \( q \) are **True**:
- \( p \wedge q \) is **True**.
- When \( p \) is **True** and \( q \) is **False**:
- \( p \wedge q \) is **False**.
- When \( p \) is **False** and \( q \) is **True**:
- \( p \wedge q \) is **False**.
- When both \( p \) and \( q \) are **False**:
- \( p \wedge q \) is **False**.

4. **Conclusion**:
From the truth table, we can conclude that \( p \wedge q \) is true **only when both \( p \) and \( q \) are true**.

### Final Answer:
\( p \wedge q \) is true when both \( p \) and \( q \) are true.

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