Definition:Set Complement

Definition
The set complement (or, when the context is established, just complement) of a set $S$ in a universe $\mathbb U$ is defined as:


 * $\complement \left ({S}\right) = \complement_\mathbb U \left ({S}\right) = \mathbb U \setminus S$

See the definition of Relative Complement for the definition of $\complement_\mathbb U \left ({S}\right)$.

Thus the complement of a set $S$ is the relative complement of $S$ in the universe, or the complement of $S$ relative to the universe.

A common alternative to the symbology $\complement \left ({S}\right)$, which we will sometimes use, is $\overline S$.

Also denoted as
No standard symbol for this concept has evolved.

Alternative notations for $\complement \left({S}\right)$ include variants of the $\complement$:
 * $\mathcal C \left({S}\right)$
 * $c \left({S}\right)$
 * $C \left({S}\right)$
 * $\mathrm C \left({S}\right)$

... and sometimes the brackets are omitted:
 * $C S$

Alternative symbols for $\overline S$ are sometimes encountered:
 * $S'$ (but it can be argued that the symbol $'$ is already overused)
 * $S^*$
 * $- S$
 * $\tilde S$
 * $\sim S$

You may encounter others.

Some authors use $S^c$ or $S^\complement$, but those can also be confused with notation used for the group theoretical conjugate.

Also see

 * Definition:Set Difference
 * Definition:Relative Complement

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

An example of a compliment is: "You're clever." Getting this wrong indicates you're not.