complementary sets

If every sets which we are addressing in a problem is a subset of a big set, the big set which contains all sets is called a universal set. Given the universal set $X$ and a object set $A$  in the problem, a difference set $X-A$  is able to be defined. We usually express it by $A^c$ .

$A^c$  is also called a complementary set of $A$ . By connotative form,
$A^c=\left\{x | x\notin A, x\in X \right\}$ 
In general as $x\in X$  has been understood and omitted, $A^c$  becomes $\left\{x | x\notin A\right\}$ .
When the universal set is implicit, it is very convenient. Of course,
$(A^c)^c=A$
$\phi^c=X$  and $X^c=\phi$  ($\phi$  is the empty set)

The expressions of complementary sets is very famous in De Morgan's law.
$(A\cup B)^c=A^c\cap B^c$
$(A\cap B)^c=A^c\cup B^c$
If one universal set for $A$  and $B$  exists, these are true. Please try proofs. However, if the universal set of $A$ is not equal the universal set of $B$ , these are not true.