Complement

If a universal set is $\mathbb{U}$, then the complement of the set $A$, denoted by $\overline{A}$, is the set of elements from $\mathbb{U}$ that are not elements of $A$. We can write it as follows:

$$\overline{A} = \mathbb{U} \setminus A$$

By this definition, the union of a set and its complement always yields the universal set, which is:

$$A \cup \overline{A} =\mathbb{U}$$

Similarly, we can extract the following facts from the definition:

$$\begin{align*} \overline{\emptyset} &= \mathbb{U} \\ \overline{\mathbb{U}} &= \emptyset \\ \overline{\overline{A}} &= A\end{align*}$$

Furthermore, it’s important to note the following: $A\ne B \Rightarrow \overline{A} \ne \overline{B}.$