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: $$A^c=\{x: \ x\in U \ and \ x\notin A\}$$

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.

Practice exercises