Definition:Set Complement

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Definition

The set complement (or, when the context is established, just complement) of a set $S$ in a universal set $\mathbb U$ 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 universal set, or the complement of $S$ relative to the universal set.

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:

SetComplement.png


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

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

Then:

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

the set of non-negative real numbers.


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

Let the universal set $\Bbb U$ 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.


Sources

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