Definition:Relative Complement

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

Alternative notation
Alternative 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.

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

Also see

 * Set Difference
 * Set Complement

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."