Complement of a set

We will call a complementary set of $A$ , and denote it as $A^{c}$ , the set difference $(U - A)$ , $U$ being the universal set. This is: $A^{c} = {x : x \in U a n d x \notin A}$

The complementary set of $A$ is the set of the elements $x$ that satisfy $x$ belongs to $U$ , and $x$ does not belong to $A$ .

Some basic properties of the complement are:

$U^{c} = \emptyset$ and $\emptyset^{c} = U$
$A - B = A \cap B^{c}$
$(A^{c})^{c} = A$
$A \cup A^{c} = U$ and $A \cap A^{c} = \emptyset$
$(A \cup B)^{c} = A^{c} \cap B^{c}$ and $(A \cap B)^{c} = A^{c} \cup B^{c}$

Property 5 is known by the name of De Morgan's Laws.