MHB Proving (A⊕B)∩A= A-B: A Simple Guide

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The discussion centers on proving the equation (A⊕B)∩A= A-B, with participants exploring various proof methods including set algebra identities and Venn diagrams. The term "symmetric difference" is clarified, indicating that A⊕B represents the union of two sets excluding their intersection. There is some confusion regarding the use of the "+" symbol to denote a difference, which is addressed by explaining that it reflects the union of mutual differences. The conversation emphasizes the importance of understanding these definitions in set theory. Overall, the thread highlights the nuances in set operations and their representations.
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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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