To solve this problem, let's first understand the concept of a contrapositive. In logic, the contrapositive of a given statement is formed by negating both the hypothesis and conclusion of the statement and then swapping them.
Given statement:
"If the lights are off, there is no one inside."
In this statement:
- Hypothesis (H): The lights are off.
- Conclusion (C): There is no one inside.
To form the contrapositive, follow these steps:
1. Negate the Conclusion:
- Original Conclusion: There is no one inside.
- Negated Conclusion: There is someone inside.
2. Negate the Hypothesis:
- Original Hypothesis: The lights are off.
- Negated Hypothesis: The lights are not off (The lights are on).
3. Swap the Negated Hypothesis and Conclusion:
- If there is someone inside (Negated Conclusion), the lights are on (Negated Hypothesis).
Therefore, the contrapositive of the statement "If the lights are off, there is no one inside" is:
"If there is someone inside, the lights are on."
Now, let's match this with the provided multiple-choice options:
1. If the lights are not off, there is someone inside.
2. If there is no one inside, the lights are not on.
3. If the lights are on, there is someone inside.
The correct option is:
"If there is someone inside, the lights are on."
