Proving (A⊕B)∩A= A-B: A Simple Guide

[putiiik]

Prove that (A⊕B)∩A= A-B! Thank you!

 

[Evgeny.Makarov]

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?

 

[HOI]

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?

 

[Evgeny.Makarov]

I believe that $A\oplus B$ denotes the symmetric difference of $A$ and $B$.

 

[HOI]

It seems very strange to us a "+" symbol to mean a "difference".

 

[I like Serena]

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.