What is the contrapositive of the following statement?

"If the lights are on, there is someone inside."

 If the lights are not on, there is no one inside.
 If there is someone inside, the lights are not off.
 If the lights are off, there is no one inside.
 If there is no one inside, the lights are off.

To find the contrapositive of a given statement, we negate both the hypothesis and the conclusion and then switch their positions.

### Given Statement:
- **Original Statement:** "If the lights are on, there is someone inside."
  - Hypothesis (P): "The lights are on."
  - Conclusion (Q): "There is someone inside."

### Contrapositive:
The contrapositive of a statement "If P, then Q" is "If not Q, then not P."

- Negating the conclusion (Q): "There is no one inside."
- Negating the hypothesis (P): "The lights are off."

### Contrapositive Form:
- "If there is no one inside, the lights are off."

### Answer:
**If there is no one inside, the lights are off.**

Would you like more details or have any questions?

### Related Questions:
1. What is the converse of the statement "If the lights are on, there is someone inside"?
2. What is the inverse of the statement "If the lights are on, there is someone inside"?
3. How does the contrapositive relate to the original statement in terms of truth value?
4. Can you provide an example of a statement where the contrapositive is easier to understand than the original?
5. How would you determine the contrapositive of a mathematical theorem?

**Tip:** Remember that the contrapositive of a statement always has the same truth value as the original statement.

What is the contrapositive of the following statement? 'If the lights are on, there is someone inside.'

Learn how to find the contrapositive of the logical statement 'If the lights are on, there is someone inside.' This problem covers fundamental concepts in logic, including conditional statements and the law of contraposition. Suitable for students in Grades 8-10.

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