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

An alternative notation for $\complement_S \left({T}\right)$ is $\mathcal C_S \left({T}\right)$. Some authors use merely $C T$ for the relative complement of $T$, it being implicit in their notation what the superset of $T$ is at the point where this notation is used.

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

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."