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:

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


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.


Different notations for $\complement_S \left({T}\right)$ include 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.


Also see

  • Results about Relative Complement can be found here.


Linguistic Note

Beware the spelling of complement. If you spell it compliment it means something completely different.

An example of a relative compliment is: "Auntie thinks you're clever."


Sources

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