Proving (A⊕B)∩A= A-B: A Simple Guide

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Discussion Overview

The discussion revolves around proving the set equality (A⊕B)∩A= A-B, focusing on different methods of proof and the definitions involved, particularly concerning the symmetric difference of sets. The scope includes theoretical aspects of set algebra and definitions.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests a proof of the set equality (A⊕B)∩A= A-B.
  • Another participant suggests multiple methods for proving the statement, including fundamental identities of set algebra and Euler-Venn diagrams, and inquires about the methods used in the original poster's course.
  • A participant questions the definition of the "direct sum" A⊕B, seeking clarification on its meaning in the context of set theory.
  • One participant asserts that A⊕B refers to the symmetric difference of sets A and B.
  • Some participants express confusion regarding the use of the "+" symbol to denote a "difference," suggesting it is unusual.
  • Another participant elaborates that the symmetric difference is defined as the union of the two sets excluding their intersection, which they argue justifies the use of the "+" symbol.

Areas of Agreement / Disagreement

Participants express differing views on the notation and definition of the symmetric difference, with some finding the "+" symbol appropriate while others find it strange. The discussion remains unresolved regarding the preferred method of proof and the clarity of definitions.

Contextual Notes

The discussion highlights potential ambiguities in the definition of the symmetric difference and the notation used, which may depend on specific contexts or conventions in set theory.

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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