Definition:Equality of Sets

Definition
Let $U$ be a universe.

Let $X, Y \in U$ be two sets.

The two sets are said to be equal iff:
 * $\forall u \in X \cup Y$,
 * $u \in X \iff u \in Y$

If the two sets are equal one writes
 * $X = Y$.

In standard Boolean logic one writes
 * $X \neq Y$

if $X$ and $Y$ are not equal.