State whether the given statements are true or false
`p implies q` is false when p and q have different truth values

Answer

To determine whether the statement "p implies q is false when p and q have different truth values" is true or false, we can analyze it using a truth table.

### Step-by-Step Solution:

1. **Understanding Implication**: The implication "p implies q" (denoted as \( p \rightarrow q \)) is defined as follows:
- It is **true** when:
- Both \( p \) and \( q \) are true (T → T = T)
- \( p \) is false and \( q \) is true (F → T = T)
- \( p \) is false and \( q \) is false (F → F = T)
- It is **false** only when:
- \( p \) is true and \( q \) is false (T → F = F)

2. **Constructing the Truth Table**: We will create a truth table for \( p \) and \( q \) to analyze the implication:

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

3. **Analyzing Different Truth Values**: Now, we will focus on the cases where \( p \) and \( q \) have different truth values:
- Case 1: \( p = T \) and \( q = F \) → \( p \rightarrow q = F \)
- Case 2: \( p = F \) and \( q = T \) → \( p \rightarrow q = T \)

4. **Conclusion**: The statement claims that "p implies q is false when p and q have different truth values". From our analysis:
- In the first case (T, F), the implication is false.
- In the second case (F, T), the implication is true.

Therefore, the statement is **false** because \( p \rightarrow q \) is not always false when \( p \) and \( q \) have different truth values.

### Final Answer:
The statement "p implies q is false when p and q have different truth values" is **False**.