Definition:Set Complement

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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 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 $\complement \left({S}\right)$ 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)
$- S$
$\tilde S$
$\sim S$

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

  • Results about set complements can be found here.

Linguistic Note

The word complement comes from the idea of complete-ment, it being the thing needed to complete something else.

It is a common mistake to confuse the words complement and compliment. Usually the latter is mistakenly used when the former is meant.

Historical Note

The concept of set complement, or logical negation, was stated by Leibniz in his initial conception of symbolic logic.