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 $\complement_S \left({T}\right)$, and is called the relative complement of $T$ in $S$, or the complement of $T$ relative to $S$.

Thus:

$\complement_S \left({T}\right) = \left\{{x \in S : x \notin T}\right\}$


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 $\complement_S \left({T}\right)$ mainly consist of variants of the $\complement$:

$\mathcal C_S \left({T}\right)$
$c_S \left({T}\right)$
$C_S \left({T}\right)$
$\mathrm C_S \left({T}\right)$

or sometimes:

$T\,^c \left({S}\right)$
$T\,^\complement \left({S}\right)$

... and sometimes the brackets are omitted:

$C_S T$


If the superset $S$ is implicit, then it can be omitted: $\complement \left({T}\right)$ 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:

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


Also see


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