Definition:Relative Complement

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."