Finding if two sets are equal I have an formula 
$(A\cap C) \cup ( B \cap  C) = C - ( A \cap B)$
and i have to prove if this equation is correct.
I transformed it into
$(\forall x\in U: x \in A \wedge x\in C) \vee ( x\in B\wedge x\in C) = x \in C \wedge ( x \notin A \wedge  x\notin B)  $
how could i further transform the equations in order to find if its true or not?
 A: A Venn diagramm might help here:

The sets do not overlap for $A \cap B \cap C$ and $C \setminus (A \cup B)$.
A: I don't think they're equal. Take $x$ such that $x$ belongs to all three sets $A,B,C$. Then $x$ is in the LHS but not the RHS.
A: Instead of transforming the equation just go after the english meaning: those  in A and C or B and C are those in C but not both A and B. This is obvioyusly not true: Consider A=B
A: 
I have an formula 
$(A\cap C) \cup ( B \cap  C) = C - ( A \cap B)$
and i have to prove if this equation is correct. I transformed it into
$(\forall x\in U: x \in A \wedge x\in C) \vee ( x\in B\wedge x\in C) = x \in C \wedge ( x \notin A \wedge  x\notin B)  $

Okay, firstly, this is a mishmash of concepts.   You don't want to use predicate logic, you want to reach for your set builder notation toolkit.   Related concepts to be sure, but not the same at all.
$$\{x\in U\mid (x\in A\wedge x\in C)\vee(x\in B\wedge x\in C)\}~=~\{x\in U\mid x\in C\wedge (x\notin (A\cap B)\}$$
Secondly, recall the deMorgan's Set Complement rules: $(A\cap B)^\complement = A^\complement\cup B^\complement$
$$\{x\in U\mid (x\in A\wedge x\in C)\vee(x\in B\wedge x\in C)\}~=~\{x\in U\mid x\in C\wedge (x\notin A\vee x\notin B)\}$$
Now just apply distribution to one side or the other to get them both in CNF or DNF form, as you prefer.   Then the verity of the statement should be apparent.
