Students are asked to prove the following statement: **Theorem:** If \( A \cap B = A \cup B \), then \( \mathcal{P}(A \cup B) = \mathcal{P}(A) \). One student provides the following proof: __________________________________________________________ **Proof:** Let \( A, B \) be sets such that \( A \cap B = A \cup B \). Let \( x \subseteq \mathcal{P}(A \cup B) \). So \( x \subseteq A \cup B \) by the definition of power sets. So \( x \subseteq A \cap B \) by our assumption. So \( x \subseteq A \) and \( x \subseteq B \) by the definition of intersection. In particular \( x \subseteq A \) by specialization. Thus, \( x \subseteq \mathcal{P}(A) \) by the definition of power sets. Thus, \( \mathcal{P}(A) = \mathcal{P}(A \cup B) \). __________________________________________________________ Is the student's proof correct? If the student is correct, then fill in the needed to - make the proof completely correct - make sure each assertion made is fully justified - make the proof written in such a way that a student in the class could follow the logic and be fully convinced that the theorem is true. If the student is incorrect, then - Identify any errors in the proof above. - Explain each error. Your explanation should be written to the student who made the error, and should try to help the student understand why what they wrote is incorrect.
Students are asked to prove the following statement: **Theorem:** If \( A \cap B = A \cup B \), then \( \mathcal{P}(A \cup B) = \mathcal{P}(A) \). One student provides the following proof: __________________________________________________________ **Proof:** Let \( A, B \) be sets such that \( A \cap B = A \cup B \). Let \( x \subseteq \mathcal{P}(A \cup B) \). So \( x \subseteq A \cup B \) by the definition of power sets. So \( x \subseteq A \cap B \) by our assumption. So \( x \subseteq A \) and \( x \subseteq B \) by the definition of intersection. In particular \( x \subseteq A \) by specialization. Thus, \( x \subseteq \mathcal{P}(A) \) by the definition of power sets. Thus, \( \mathcal{P}(A) = \mathcal{P}(A \cup B) \). __________________________________________________________ Is the student's proof correct? If the student is correct, then fill in the needed to - make the proof completely correct - make sure each assertion made is fully justified - make the proof written in such a way that a student in the class could follow the logic and be fully convinced that the theorem is true. If the student is incorrect, then - Identify any errors in the proof above. - Explain each error. Your explanation should be written to the student who made the error, and should try to help the student understand why what they wrote is incorrect.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:Students are asked to prove the following statement:
**Theorem:** If \( A \cap B = A \cup B \), then \( \mathcal{P}(A \cup B) = \mathcal{P}(A) \).
One student provides the following proof:
__________________________________________________________
**Proof:**
Let \( A, B \) be sets such that \( A \cap B = A \cup B \).
Let \( x \subseteq \mathcal{P}(A \cup B) \).
So \( x \subseteq A \cup B \) by the definition of power sets.
So \( x \subseteq A \cap B \) by our assumption.
So \( x \subseteq A \) and \( x \subseteq B \) by the definition of intersection.
In particular \( x \subseteq A \) by specialization.
Thus, \( x \subseteq \mathcal{P}(A) \) by the definition of power sets.
Thus, \( \mathcal{P}(A) = \mathcal{P}(A \cup B) \).
__________________________________________________________
Is the student's proof correct?
If the student is correct, then fill in the needed to
- make the proof completely correct
- make sure each assertion made is fully justified
- make the proof written in such a way that a student in the class could follow the logic and be fully convinced that the theorem is true.
If the student is incorrect, then
- Identify any errors in the proof above.
- Explain each error. Your explanation should be written to the student who made the error, and should try to help the student understand why what they wrote is incorrect.
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