When does the inverse of the statement `~ p Rightarrow q ` results in T ?

Options

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

Answer

To determine when the inverse of the statement \( \sim p \Rightarrow q \) results in true, we first need to understand the components involved in this logical expression.

### Step-by-Step Solution:

1. **Understanding the Inverse**:
The inverse of the statement \( \sim p \Rightarrow q \) is \( p \Rightarrow \sim q \). This means we need to analyze when the statement \( p \Rightarrow \sim q \) is true.

2. **Truth Table for Implication**:
Recall that an implication \( A \Rightarrow B \) is only false when \( A \) is true and \( B \) is false. Therefore, we need to analyze the truth values of \( p \) and \( \sim q \).

3. **Constructing the Truth Table**:
We will create a truth table for \( p \), \( q \), \( \sim q \), and \( p \Rightarrow \sim q \):

| \( p \) | \( q \) | \( \sim q \) | \( p \Rightarrow \sim q \) |
|---------|---------|---------------|-----------------------------|
| T | T | F | F |
| T | F | T | T |
| F | T | F | T |
| F | F | T | T |

4. **Analyzing the Table**:
- When \( p \) is true and \( q \) is true, \( \sim q \) is false, so \( p \Rightarrow \sim q \) is false.
- When \( p \) is true and \( q \) is false, \( \sim q \) is true, so \( p \Rightarrow \sim q \) is true.
- When \( p \) is false and \( q \) is true, \( \sim q \) is false, so \( p \Rightarrow \sim q \) is true.
- When \( p \) is false and \( q \) is false, \( \sim q \) is true, so \( p \Rightarrow \sim q \) is true.

5. **Conclusion**:
The inverse \( p \Rightarrow \sim q \) results in true in the following cases:
- When \( p \) is true and \( q \) is false.
- When \( p \) is false and \( q \) is true.
- When \( p \) is false and \( q \) is false.

Thus, the inverse \( \sim p \Rightarrow q \) results in true when either \( p \) is false or \( q \) is false.