Simple set exercise seems not so simple Exercise about sets from Birkhoff's "Modern Applied Algebra".

Prove that for operation $\ \Delta $ , defined as

$\ R \Delta S = (R \cap S^c) \cup (R^c \cap S) $

following is true:

$\ R \Delta ( S \Delta T ) = ( R \Delta S ) \Delta T $


($\ S^c $ is complement of  $\ S $)
It's meant to be very simple, being placed in the first excercise block of the book. When I started to expand both sides of equations in order to prove that they're equal, I got this monster just for the left side:

$\ R \Delta ( S \Delta T ) = \Bigl( R \cap \bigl( (S \cap T^c) \cup (S^c \cap T) \bigr)^c \Bigr) \cup \Bigl(R^c \cap \bigl( (S \cap T^c) \cup (S^c \cap T) \bigr) \Bigr) $

For the right:

$\ ( R \Delta S ) \Delta T = \Bigl(\bigl( (R \cap S^c) \cup (R^c \cap S) \bigr) \cap T^c  \Bigr) \cup \Bigl( \bigl( (R \cap S^c) \cup (R^c \cap S) \bigr)^c  \cap T \Bigr)   $

I've tried to simplify this expression, tried to somehow rearrange it, but no luck. Am I going the wrong way? Or what should I do with what I have?

This is the symmetric difference. It includes all elements that are in exactly one of the two sets. In binary terms, it's the XOR operation. Independent of the order of the operations on the three sets, the result will contain exactly the elements that are in an odd number of the sets.
