# SETS problem

this is one problem which i dont 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, thats the closet symbol i could find
the oplus symbol is a circle with one line going across it and one line going down

## Answers and Replies

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

http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php [Broken]

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StatusX
Homework Helper
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?

yes that is the symbol which im referring to...symmetric difference

i dont know where to start..i have no clue on what to do?

StatusX
Homework Helper
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.

i have tried to draw circles before...but i just keep getting lost..i dont 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
Homework Helper
Ok, then just write out the definitions. Get everything in terms of unions, intersections and complements, which should be easier to work with.

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

HallsofIvy
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.