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:


 * $\map \complement S = \relcomp {\mathbb U} S = \mathbb U \setminus S$

See the definition of Relative Complement for the definition of $\relcomp {\mathbb U} S$.

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 $\map \complement S$, which we will sometimes use, is $\overline S$.

Also known as
Some sources use the term absolute complement, in apposition to relative complement.

No standard symbol for this concept has evolved.

Alternative notations for $\map \complement S$ include variants of the $\complement$:
 * $\map {\CC} S$
 * $\map c S$
 * $\map C S$
 * $\map {\operatorname C} S$
 * $\map {\operatorname {\mathbf C} } S$
 * ${}_c S$

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