Question regarding algebra of sets

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SUMMARY

The discussion focuses on proving that the operation of Symmetric Difference imposes a group structure on the power set 2^X of a set X. The key equation referenced is A - B = (A ∪ B) - B. The proof involves demonstrating the implications of elements belonging to A - B and A ∪ B - B, confirming the group structure through logical deductions. The consensus among participants is that the proof appears correct and well-structured.

PREREQUISITES
  • Understanding of set theory concepts, specifically Symmetric Difference
  • Familiarity with power sets, denoted as 2^X
  • Knowledge of logical implications and proof techniques in mathematics
  • Basic operations on sets, including union (∪) and difference (-)
NEXT STEPS
  • Study the properties of Symmetric Difference in set theory
  • Explore group theory fundamentals and how they apply to set operations
  • Investigate the implications of power sets in algebraic structures
  • Review logical proof techniques to strengthen mathematical argumentation
USEFUL FOR

Students in introductory algebra courses, mathematicians interested in set theory, and educators teaching proof techniques in mathematics.

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Homework Statement



I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on however...

I've worked out a proof, and I can see it relies heavily on this equation:

Let A, B be sets.

i. A - B = (AuB) - B

Homework Equations



The Attempt at a Solution



My proof goes like this: (=>)Let x in A - B. Then, x is in A, implying x is in AuB. Since x cannot be in B, then it also falls in AuB - B.
(<=): Let x in AuB - B. Then x cannot be in B, and must therefore be in AuB. x not in B but still in AuB - B implies that x must be in A and not in B, which means it also falls in A - B.

Did I miss anything/ any mistakes?
 
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ssayani87 said:

Homework Statement



I'm working on a proof for Intro. to Algebra, and the problem deals with proving that the operation of Symmetric Difference imposes group structure on the power set 2^x (set of all subsets) of a given set X. There is a part of my proof that I'd like to get some advice on however...

I've worked out a proof, and I can see it relies heavily on this equation:

Let A, B be sets.

i. A - B = (AuB) - B

Homework Equations



The Attempt at a Solution



My proof goes like this: (=>)Let x in A - B. Then, x is in A, implying x is in AuB. Since x cannot be in B, then it also falls in AuB - B.
(<=): Let x in AuB - B. Then x cannot be in B, and must therefore be in AuB. x not in B but still in AuB - B implies that x must be in A and not in B, which means it also falls in A - B.

Did I miss anything/ any mistakes?

Sounds fine to me.
 

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