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?