Question
Consider the statement, If you make a mess in the bathroom, then I will be upset with you.
If this statement is true, which of the following statements is false?
If I am upset with you, then you made a mess in the bathroom.
If I am not upset with you, then you did not make a mess in the bathroom.
If you don’t make a mess in the bathroom, then I will not be upset with you.
You made a mess in the bathroom and I am not upset with you.
If this statement is true, which of the following statements is false?
If I am upset with you, then you made a mess in the bathroom.
If I am not upset with you, then you did not make a mess in the bathroom.
If you don’t make a mess in the bathroom, then I will not be upset with you.
You made a mess in the bathroom and I am not upset with you.
Answers
Answered by
GPT-4o mini
The original statement "If you make a mess in the bathroom, then I will be upset with you" can be represented as:
- Let \( P \) = you make a mess in the bathroom.
- Let \( Q \) = I will be upset with you.
So, the statement can be written as \( P \implies Q \) (if \( P \) then \( Q \)).
We need to evaluate the truth of the four statements provided, given that the original implication is true.
1. **If I am upset with you, then you made a mess in the bathroom.**
- This is the converse of the original statement: \( Q \implies P \). The truth of the original statement does not guarantee the truth of the converse, so this might be false.
2. **If I am not upset with you, then you did not make a mess in the bathroom.**
- This can be rewritten using contraposition: If \( \neg Q \) then \( \neg P \) (if I am not upset, then you did not make a mess). The contrapositive of a true statement is true. Hence, this statement is true.
3. **If you don’t make a mess in the bathroom, then I will not be upset with you.**
- This can also be rewritten using contraposition: If \( \neg P \) then \( \neg Q \). Again, the contrapositive of a true statement is true. Therefore, this statement is true.
4. **You made a mess in the bathroom and I am not upset with you.**
- This statement asserts that \( P \) is true (you made a mess) and \( Q \) is false (I am not upset). This is directly contradictory to the initial statement, which asserts that if \( P \) is true, then \( Q \) must also be true. Thus, this statement is false.
Upon review, the statement that is false is:
**You made a mess in the bathroom and I am not upset with you.**
- Let \( P \) = you make a mess in the bathroom.
- Let \( Q \) = I will be upset with you.
So, the statement can be written as \( P \implies Q \) (if \( P \) then \( Q \)).
We need to evaluate the truth of the four statements provided, given that the original implication is true.
1. **If I am upset with you, then you made a mess in the bathroom.**
- This is the converse of the original statement: \( Q \implies P \). The truth of the original statement does not guarantee the truth of the converse, so this might be false.
2. **If I am not upset with you, then you did not make a mess in the bathroom.**
- This can be rewritten using contraposition: If \( \neg Q \) then \( \neg P \) (if I am not upset, then you did not make a mess). The contrapositive of a true statement is true. Hence, this statement is true.
3. **If you don’t make a mess in the bathroom, then I will not be upset with you.**
- This can also be rewritten using contraposition: If \( \neg P \) then \( \neg Q \). Again, the contrapositive of a true statement is true. Therefore, this statement is true.
4. **You made a mess in the bathroom and I am not upset with you.**
- This statement asserts that \( P \) is true (you made a mess) and \( Q \) is false (I am not upset). This is directly contradictory to the initial statement, which asserts that if \( P \) is true, then \( Q \) must also be true. Thus, this statement is false.
Upon review, the statement that is false is:
**You made a mess in the bathroom and I am not upset with you.**
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