Proving the Oplus Symbol: A, B, C, D Sets - Solving the SETS Problem

[majeedh]

this is one problem which i don't know how to start..im not sure which proof i should use to solve this problem
the problem is:
If A,B,C,and D are sets, does it follow
(A Φ B) Φ (C Φ D) = (A Φ C) Φ (B Φ D)
the symbol that is separting the characters is called the oplus symbol, that's the closet symbol i could find
the oplus symbol is a circle with one line going across it and one line going down

 

[quasar987]

Does A Φ B stands for the disjoint union of A and B?

http://en.wikipedia.org/wiki/Disjoint_union

 

[majeedh]

no it doesnt
the symbol I am trying to illustrate there is the oplus which can be viewed at this link

http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php

 

[StatusX]

Does that mean the XOR operation, ie, $A \oplus B = (A-B) \cup (B-A)$, also known as the symmetric difference? What have you tried?

 

[majeedh]

yes that is the symbol which I am referring to...symmetric difference

i don't know where to start..i have no clue on what to do?

 

[StatusX]

Start by trying to see if it's true. You can draw a picture, something like 4 overlapping circles, and see if you can find the regions corresponding to each side of that equation.

 

[majeedh]

i have tried to draw circles before...but i just keep getting lost..i don't know how to proof this and which would be the best proof to use in such a case..im totally lost on how to start the problem

 

[StatusX]

Ok, then just write out the definitions. Get everything in terms of unions, intersections and complements, which should be easier to work with.

 

[majeedh]

im going to see what i can come up with..but iam clueless in what to do...

 

[HallsofIvy]

You want to prove that two sets, the set on the left side and the set on the right side, are equal. Typically you do that by proving that anything in one of the sets is also in the other.

Suppose x is in $(A\oplus B)\oplus (C\oplus D)$. What must be true about it? From the definition, if a point, p, is in $A\oplus B$ is it in A or B? exactly what is true about p from the definitions? Once you know exactly where, relative to A, B, C, D, p must be in order to be in the left set, use that information to show that it must be in the right set.

Now reverse and do it the other way.