If `p and q` are two statement then `(p <-> ~ q)` is true when : (a) `p and q` both are true (b) `p and q` both are false (c) `p` is false and `q` ia true (d) Non of these

Options

A. p and q both are true
B. both p and q are false
C. p is true and q is false
D. none of these

Answer

To determine when the statement \( (p \leftrightarrow \neg q) \) is true, we will construct a truth table for the variables \( p \) and \( q \), and analyze the implications.

### Step 1: Construct the Truth Table

1. **List all possible truth values for \( p \) and \( q \)**:
- \( p \) can be True (T) or False (F).
- \( q \) can be True (T) or False (F).

2. **Create the columns for \( p \), \( q \), and \( \neg q \)**:
- \( \neg q \) is the negation of \( q \).

3. **Determine the values for \( p \leftrightarrow \neg q \)**:
- The biconditional \( p \leftrightarrow \neg q \) is true if both \( p \) and \( \neg q \) have the same truth value.

The truth table will look like this:

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

### Step 2: Analyze the Truth Table

From the truth table, we can see when \( p \leftrightarrow \neg q \) is true:
- When \( p \) is True and \( q \) is False (2nd row).
- When \( p \) is False and \( q \) is True (3rd row).

### Step 3: Evaluate the Options

Now, let's evaluate the options provided in the question:

- (a) \( p \) and \( q \) both are true: **False** (1st row).
- (b) \( p \) and \( q \) both are false: **False** (4th row).
- (c) \( p \) is false and \( q \) is true: **True** (3rd row).
- (d) None of these: **False**.

### Conclusion

The correct answer is **(c) \( p \) is false and \( q \) is true**.

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