Complement¶

Definition¶

Let $A\subseteq U\text{,}$ where $U$ is a fixed universal set. The complement of $A$ relative to $U$ is

A^{\complement}=U\setminus A=\{x\in U\colon x\notin A\}.

The complement is the difference set $U\setminus A$ once the universal set $U$ has been fixed.

Interpretation¶

The complement contains everything in the chosen universe that does not belong to the set. It therefore depends on which universal set is being used: for $A=\{1,2\}$ ${1, 2}$ , the complement is $\{3,4\}$ when $U=\{1,2,3,4\}$ ${1, 2, 3, 4}$ , but $\{3\}$ when $U=\{1,2,3\}$ ${1, 2, 3}$ .

A rectangle for the universal set U with a disc for the set A; the shaded region outside the disc is the complement of A.

The rectangle is $U$ and the disc is $A$ ; the shaded region outside the disc is $A^{\complement}$ . The disc and the shaded region do not overlap and together fill the rectangle, which is $A\cap A^{\complement}=\varnothing$ and $A\cup A^{\complement}=U$ $U$ ; taking the complement of the shaded region gives back the disc.

In logical terms, the complement is "not": $x\in A^{\complement}$ means $x\in U$ and $x\notin A$ $A$ . A common slip is to distribute the complement without switching the operation: with $U=\{1,2,3,4\}$ ${1, 2, 3, 4}$ , $A=\{1,3\}$ and $B=\{1,2\}$ ${1, 2}$ , we get $(A\cup B)^{\complement}=\{4\}$ ${4}$ , whereas $A^{\complement}\cup B^{\complement}=\{2,3,4\}$ ${2, 3, 4}$ . De Morgan's laws give the correct rule.

The idea returns in probability as the complement of an event: the event that a given event does not happen.

Key Results¶

$(A^{\complement})^{\complement}=A\text{.}$
$U^{\complement}=\varnothing$ and $\varnothing^{\complement}=U\text{.}$
De Morgan's laws:

$\begin{aligned} (A\cup B)^{\complement}&=A^{\complement}\cap B^{\complement},\\ (A\cap B)^{\complement}&=A^{\complement}\cup B^{\complement}. \end{aligned}$

Examples¶

If $U=\{1,2,3,4\}$ and $A=\{1,3\}\text{,}$ then $A^{\complement}=\{2,4\}\text{.}$
If $U=\mathbb{R}$ and $A=[0,\infty)\text{,}$ then $A^{\complement}=(-\infty,0)\text{.}$

Prerequisites¶

Sources¶

Math I: Calculus · Lecture 1
Math II: Linear Algebra · Week 1 (set-theory recall)

Aliases: set complement, absolute complement Symbols: element of, infinity, intersection, real numbers, subset, union

Last updated: 20 October 2026 03:02