Proving (A⊕B)∩A= A-B: A Simple Guide

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SUMMARY

The discussion centers on proving the equation (A⊕B)∩A= A-B, where A⊕B represents the symmetric difference of sets A and B. Participants highlight various proof methods, including fundamental identities of set algebra and Euler-Venn diagrams. The symmetric difference is defined as the union of both sets excluding their intersection, justifying the use of the "+" symbol in this context. The conversation emphasizes the importance of understanding these definitions and identities in set theory.

PREREQUISITES
  • Fundamental identities of set algebra
  • Understanding of symmetric difference in set theory
  • Familiarity with Euler-Venn diagrams
  • Basic definitions of set operations: union, intersection, and difference
NEXT STEPS
  • Study the fundamental identities of set algebra in detail
  • Learn how to construct and interpret Euler-Venn diagrams
  • Explore the properties and applications of symmetric difference in set theory
  • Review definitions and examples of set operations: union, intersection, and difference
USEFUL FOR

Students of mathematics, educators teaching set theory, and anyone interested in understanding the properties and proofs related to set operations.

putiiik
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Prove that (A⊕B)∩A= A-B! Thank you!
 
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There are various ways of proving this: using the fundamental identities of set algebra, using Euler-Venn diagrams, by definition using mutual inclusion of the left- and right-hand sides, etc. Which one is used in your course? And if it is the first method, are you familiar with the fundamental identities?
 
For arbitrary sets, union $A\cup B$, intersection $A\cap B$, and difference A\ B, are defined but how are you defining the "direct sum" $A\bigoplus B$ of sets?
 
It seems very strange to us a "+" symbol to mean a "difference".
 
Country Boy said:
It seems very strange to us a "+" symbol to mean a "difference".
It's the union of both sets except for their intersection.
As such a "+" seems appropriate.
It's just that to define it, we typically take the union of the 2 mutual differences, which is apparently why it is called symmetric difference.
 

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