# Definition:Set Complement

## Definition

The set complement (or, when the context is established, just complement) of a set $S$ in a universe $\mathbb U$ is defined as:

$\map \complement S = \relcomp {\mathbb U} S = \mathbb U \setminus S$

See the definition of Relative Complement for the definition of $\relcomp {\mathbb U} S$.

Thus the complement of a set $S$ is the relative complement of $S$ in the universe, or the complement of $S$ relative to the universe.

A common alternative to the symbology $\map \complement S$, which we will sometimes use, is $\overline S$.

## Illustration by Venn Diagram

The complement $\map \complement T$ of the set $T$ with respect to the universal set $\mathbb U$ is illustrated in the following Venn diagram by the coloured area:

## Notation

No standard symbol for the concept of set complement has evolved.

Alternative notations for $\map \complement S$ include variants of the $\complement$:

$\map {\CC} S$
$\map c S$
$\map C S$
$\map {\operatorname C} S$
$\map {\operatorname {\mathbf C} } S$
${}_c S$

and sometimes the brackets are omitted:

$C S$

Alternative symbols for $\overline S$ are sometimes encountered:

$S'$ (but it can be argued that the symbol $'$ is already overused)
$S^*$
$- S$
$\tilde S$
$\sim S$

You may encounter others.

Some authors use $S^c$ or $S^\complement$, but those can also be confused with notation used for the group theoretical conjugate.

## Also known as

Some older sources use the term absolute complement, in apposition to relative complement.

## Examples

### $\R_{>0}$ in $\R$

Let the universe $\Bbb U$ be defined to be the set of real numbers $\R$.

Let the set of (strictly) positive real numbers be denoted by $\R_{>0}$.

Then:

$\relcomp {} {\R_{>0} } = \R_{\le 0}$

### $\R_{>0}$ in $\C$

Let the universe $\Bbb U$ be defined to be the set of real numbers $\C$.

Let the set of (strictly) positive real numbers be denoted by $\R_{>0}$.

Then:

$\relcomp {} {\R_{>0} } = \set {x + i y: y \ne 0 \text { or } x \le 0}$

## Also see

• Results about set complements can be found here.

## Historical Note

The concept of set complement, or logical negation, was stated by Leibniz in his initial conception of symbolic logic.

## Linguistic Note

The word complement comes from the idea of complete-ment, it being the thing needed to complete something else.

It is a common mistake to confuse the words complement and compliment.

Usually the latter is mistakenly used when the former is meant.