In which of the following is equivalent cases, ` p Rightarrow q ` is false ?

Options

A. p is true, q is true
B. p is false, q is true
C. p is true, q is false
D. none of these

Answer

To determine when the implication \( p \Rightarrow q \) is false, we can analyze the truth values of \( p \) and \( q \). The implication \( p \Rightarrow q \) is defined as follows:

- It is true if both \( p \) and \( q \) are true.
- It is true if \( p \) is false (regardless of \( q \)).
- It is false only when \( p \) is true and \( q \) is false.

### Step-by-step Solution:

1. **Understanding the Implication**: Recall that the implication \( p \Rightarrow q \) can be interpreted as "if \( p \) is true, then \( q \) must also be true." The only scenario where this statement is false is when \( p \) is true and \( q \) is false.

2. **Constructing the Truth Table**:
- Create a truth table with all possible combinations of truth values for \( p \) and \( q \).
- The combinations are:
- \( p = \text{True}, q = \text{True} \)
- \( p = \text{True}, q = \text{False} \)
- \( p = \text{False}, q = \text{True} \)
- \( p = \text{False}, q = \text{False} \)

3. **Filling the Truth Table**:
- For each combination, determine the truth value of \( p \Rightarrow q \):
- \( p = \text{True}, q = \text{True} \) → \( p \Rightarrow q = \text{True} \)
- \( p = \text{True}, q = \text{False} \) → \( p \Rightarrow q = \text{False} \)
- \( p = \text{False}, q = \text{True} \) → \( p \Rightarrow q = \text{True} \)
- \( p = \text{False}, q = \text{False} \) → \( p \Rightarrow q = \text{True} \)

4. **Identifying When \( p \Rightarrow q \) is False**:
- From the truth table, we see that \( p \Rightarrow q \) is false only in the case where \( p \) is true and \( q \) is false.

5. **Conclusion**:
- Therefore, the only case in which \( p \Rightarrow q \) is false is when \( p \) is true and \( q \) is false.

### Final Answer:
The implication \( p \Rightarrow q \) is false when \( p \) is true and \( q \) is false.