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 see

 * Set Difference
 * Relative Complement

Notation
No standard symbol for this concept has evolved. There are alternative symbols for $\complement \left ({S}\right)$ and $\overline S$. $\mathcal C \left ({S}\right)$ is sometimes encountered, and may appear occasionally on this website. Another common one is $S'$, but it can be argued that the symbol $'$ is already overused.

Some authors use $S^c$ or $S^\complement$, but those can also be confused with notation used for the group theoretical conjugate. Some authors use $C S$. Another one is $S^*$, and another is $\tilde S$. You may encounter others.

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.