If p and q are two simple propositions, then ` p to q` is false when

Options

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

Answer

To determine when the implication \( p \to q \) (read as "p implies q") is false, we can use a truth table. Let's break down the steps to arrive at the solution.

### Step-by-Step Solution:

1. **Understanding Implication**: The implication \( p \to q \) is defined as:
- 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.

2. **Constructing the Truth Table**:
Let's create a truth table for \( p \) and \( q \) along with \( p \to q \):

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

Here, T represents true and F represents false.

3. **Analyzing the Truth Table**:
From the truth table, we can see that:
- When \( p \) is true (T) and \( q \) is true (T), \( p \to q \) is true (T).
- When \( p \) is true (T) and \( q \) is false (F), \( p \to q \) is false (F).
- When \( p \) is false (F), \( p \to q \) is true (T) regardless of whether \( q \) is true or false.

4. **Conclusion**:
The implication \( p \to q \) is false only in the case where \( p \) is true and \( q \) is false. Therefore, we conclude that \( p \to q \) is false when:
- \( p \) is true (T)
- \( q \) is false (F)

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