Definition:Set Complement

From ProofWiki

Jump to: navigation, search

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 $\overline S$.

No standard symbol for this concept has evolved.

Alternative notations for $\complement \left({T}\right)$ 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)

$S^*$

$- S$

$\tilde S$

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.

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.