Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
This is an exercise I was trying to do ( not homework ) and I got stuck as follows:
Working from $A \setminus ( B \setminus C )$:
$x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$
At this point I'm not sure how to proceed.
Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:
$ x \in ( A \setminus B) \cup ( A \cap C ) $
$ x \in ( A \setminus B) \vee x \in ( A \cap C ) $
$ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $
$ x \in A \wedge (x \notin B \vee x \notin C ) $
$ x \in A \wedge x \notin ( B \cap C ) $
$ A \setminus ( B \cap C ) $
I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-)