Definition:Relative Complement

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Definition

Let $S$ be a set, and let $T \subseteq S$, that is: let $T$ be a subset of $S$.

Then the set difference $S \setminus T$ can be written $\relcomp S T$, and is called the relative complement of $T$ in $S$, or the complement of $T$ relative to $S$.

Thus:

$\relcomp S T = \set {x \in S : x \notin T}$


Illustration by Venn Diagram

The relative complement $\complement_S \left({T}\right)$ of the set $T$ with respect to $S$ is illustrated in the following Venn diagram by the red area:

RelativeComplement.png


Also known as

Some authors call this the complement and use the term relative complement for the set difference $S \setminus T$ when the stipulation $T \subseteq S$ is not required.


Others emphasize the connection with set difference by referring to the relative complement as a proper difference.

Thus, in this view, the relative complement is a specific case of a set difference.


Different notations for $\relcomp S T$ mainly consist of variants of the $\complement$:

$\map {\mathcal C_S} T$
$\map {c_S} T$
$\map {C_S} T$
$\operatorname C_S \paren T$

or sometimes:

$\map {T\,^c} S$
$\map {T\,^\complement} S$

... and sometimes the brackets are omitted:

$C_S T$


If the superset $S$ is implicit, then it can be omitted: $\complement \paren T$ etc. See the notation for set complement.


Some sources do not bother to introduce a specific notation for the relative complement, and instead just use the various notation for set difference:

$S \setminus T$
$S / T$
$S - T$


Examples

Examples

Let $E$ denote the set of all even integers.

Then the relative complement of $E$ in $\Z$:

$\complement_\Z \paren E$

is the set of all odd integers.


Also see

  • Results about Relative Complement can be found here.


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.


Technical Note

The $\LaTeX$ code for \(\relcomp {S} {T}\) is \relcomp {S} {T} .

This is a custom construct which has been set up specifically for the convenience of the users of $\mathsf{Pr} \infty \mathsf{fWiki}$.

Note that there are two arguments to this operator: the subscript, and the part between the brackets.

If either part is a single symbol, then the braces can be omitted, for example:

\relcomp S T


Sources