Complement

Definition

The complement of a set $A$ (with respect to a universal set $U$) is the set of all elements in $U$ that are not in $A$:

\[A^c = \{x \in U : x \notin A\}\]

Symbol	Meaning
$A^c$	Complement of $A$
$\overline{A}$	Alternative notation
$U \setminus A$	Set difference with universal set

\[(A^c)^c = A\]

The complement of the complement is the original set.

\[(A \cup B)^c = A^c \cap B^c$$ $$(A \cap B)^c = A^c \cup B^c\]

Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{1, 2\}$.

Then $A^c = \{3, 4, 5\}$.

Let $U = \mathbb{R}$ and $A = \{x : x > 0\}$.

Then $A^c = \{x : x \leq 0\}$.

Backlinks

Symbol	Meaning
\(A^c\)	Complement of \(A\)
\(\overline{A}\)	Alternative notation
\(U \setminus A\)	Set difference with universal set