Suppose p and q are two statements and - p ^ q is true, then truth values of p and q are respectively

Options

A. T, T
B. T, F
C. F, T
D. F, F

Answer

To solve the problem, we need to determine the truth values of statements \( p \) and \( q \) given that the negation of \( p \) and \( q \) (denoted as \( - (p \land q) \)) is true.

### Step-by-Step Solution:

1. **Understanding the Statement**:
We know that \( - (p \land q) \) is true. This means that the statement "not (p and q)" is true.

2. **Using Logical Equivalence**:
The negation of a conjunction can be rewritten using De Morgan's laws:
\[ - (p \land q) \equiv -p \lor -q \]
This means that at least one of \( -p \) or \( -q \) must be true.

3. **Truth Table Construction**:
We will create a truth table for \( p \) and \( q \) to analyze the possible truth values.

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

4. **Analyzing the Truth Table**:
From the truth table, we see that \( - (p \land q) \) is true in the following cases:
- When \( p \) is true and \( q \) is false (2nd row).
- When \( p \) is false and \( q \) is true (3rd row).
- When both \( p \) and \( q \) are false (4th row).

5. **Finding the Required Condition**:
However, we need to find the case where \( - (p \land q) \) is true. The only scenario where \( p \land q \) is false and \( - (p \land q) \) is true is when \( p \) is false and \( q \) is true.

6. **Conclusion**:
Therefore, the truth values of \( p \) and \( q \) are:
- \( p = \text{False} \)
- \( q = \text{True} \)

### Final Answer:
The truth values of \( p \) and \( q \) are respectively \( F \) (False) and \( T \) (True).

---