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:

  1. $$U^c=\emptyset$$ and $$\emptyset^c=U$$
  2. $$A-B=A\cap B^c$$
  3. $$(A^c)^c=A$$
  4. $$A\cup A^c=U$$ and $$A\cap A^c=\emptyset$$
  5. $$(A\cup B)^c=A^c\cap B^c$$ and $$(A\cap B)^c=A^c\cup B^c$$

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