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

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 search 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:
 * $S \setminus T$

Also see

 * Definition:Set Difference
 * Definition: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 when Auntie tells you how clever you are.