- #1
Symmetric difference problem (Real Analysis)
- Thread starter phillyolly
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Homework Help Overview
The discussion revolves around understanding the symmetric difference of two sets in the context of real analysis. Participants are exploring the definitions and proofs related to set operations, particularly focusing on the symmetric difference defined as elements belonging to either set A or set B, but not both.
Discussion Character
- Conceptual clarification, Mathematical reasoning, Assumption checking
Approaches and Questions Raised
- The original poster questions the requirements of the problem, seeking clarity on whether to draw a diagram or prove specific statements. Some participants suggest drawing a Venn diagram and emphasize understanding the definitions of set operations involved. Others raise concerns about the correctness of the original poster's notation and reasoning, prompting a breakdown of the concepts into more fundamental terms.
Discussion Status
The discussion is active, with participants providing guidance on how to approach the problem. There is an emphasis on ensuring clarity in notation and understanding the definitions of set operations. Multiple interpretations of the problem are being explored, particularly regarding the formal proof of the symmetric difference.
Contextual Notes
Participants are addressing potential misunderstandings related to set notation and operations, indicating that the original poster may be new to the topic. There is a focus on ensuring that the foundational concepts are well understood before proceeding with the proof.
- #2
Raskolnikov
- 193
- 2
Draw a Venn diagram with sets A and B. Shade in the area "that belongs to either A or B, but not both."
Then, I'd say just make sure you first know exactly what (a) means, i.e., what is A\B and B\A. You should easily be able to see that the definition of D in (a) is the same as the more intuitive definition "either A or B, but not both."
Finally, you should write a quick formal proof that shows the definition in (a) is the same as that in (b).
i.e, prove that (A \ B) U (B \ A) = (A U B) \ (A [tex]\cap[/tex] B).
Since you're new to all of this, don't skip steps. Take care of the details!
Then, I'd say just make sure you first know exactly what (a) means, i.e., what is A\B and B\A. You should easily be able to see that the definition of D in (a) is the same as the more intuitive definition "either A or B, but not both."
Finally, you should write a quick formal proof that shows the definition in (a) is the same as that in (b).
i.e, prove that (A \ B) U (B \ A) = (A U B) \ (A [tex]\cap[/tex] B).
Since you're new to all of this, don't skip steps. Take care of the details!
- #3
- #4
Raskolnikov
- 193
- 2
hehe, first of all, you have A/B instead of A\B.
Secondly, your second line is utterly meaningless. The union operation "U" is only defined on sets, not on boolean statements such as "x is element of A\B."
Thirdly, [tex]x \in A \backslash x \in B[/tex] is also utterly meaningless. The relative complement operation "\" is defined only on sets.
Since you are new to all this, break this down to the very basics.
[tex]x \in (A \backslash B) \cup (B \backslash A).[/tex]
[tex]x \in (A \backslash B) \vee x \in (B \backslash A).[/tex]
[tex](x \in A \wedge x \notin B) \vee (x \in B \wedge x \notin A).[/tex]
Try completing from there. Step-by-step! I know a lot of it seems like just grunt work, but it's a very important fundamental approach :P!
Secondly, your second line is utterly meaningless. The union operation "U" is only defined on sets, not on boolean statements such as "x is element of A\B."
Thirdly, [tex]x \in A \backslash x \in B[/tex] is also utterly meaningless. The relative complement operation "\" is defined only on sets.
Since you are new to all this, break this down to the very basics.
[tex]x \in (A \backslash B) \cup (B \backslash A).[/tex]
[tex]x \in (A \backslash B) \vee x \in (B \backslash A).[/tex]
[tex](x \in A \wedge x \notin B) \vee (x \in B \wedge x \notin A).[/tex]
Try completing from there. Step-by-step! I know a lot of it seems like just grunt work, but it's a very important fundamental approach :P!
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