Complement (set theory)

A circle filled with red inside a square. The area outside the circle is unfilled. The borders of both the circle and the square are black.

If

A

is the area colored red in this image…

An unfilled circle inside a square. The area inside the square not covered by the circle is filled with red. The borders of both the circle and the square are black.

… then the complement of

A

is everything else.

In set theory, the complement of a set $A$ , often denoted by $A^{\complement }$ (or $A'$ ),^[1] is the set of elements not in $A$ .^[2]

When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set $U$ , the absolute complement of $A$ is the set of elements in $U$ that are not in $A$ .

The relative complement of $A$ with respect to a set $B$ , also termed the set difference of $B$ and $A$ , written $B\setminus A,$ is the set of elements in $B$ that are not in $A$ .

Absolute complement

Definition

If $A$ is a set, then the absolute complement of $A$ (or simply the complement of $A$ ) is the set of elements not in $A$ (within a larger set that is implicitly defined). In other words, let $U$ be a set that contains all the elements under study; if there is no need to mention $U$ , either because it has been previously specified, or it is obvious and unique, then the absolute complement of $A$ is the relative complement of $A$ in $U$ :^[3] $A^{\complement }=U\setminus A=\{x\in U:x\notin A\}.$

The absolute complement of $A$ is usually denoted by $A^{\complement }$ . Other notations include ${\overline {A}},A',$ ^[2] $\complement _{U}A,{\text{ and }}\complement A.$ ^[4]

Examples

Assume that the universe is the set of integers. If $A$ is the set of odd numbers, then the complement of $A$ is the set of even numbers. If $B$ is the set of multiples of 3, then the complement of $B$ is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
Assume that the universe is the standard 52-card deck. If the set $A$ is the suit of spades, then the complement of $A$ is the union of the suits of clubs, diamonds, and hearts. If the set $B$ is the union of the suits of clubs and diamonds, then the complement of $B$ is the union of the suits of hearts and spades.
When the universe is the universe of sets described in formalized set theory, the absolute complement of a set is generally not itself a set, but rather a proper class. For more info, see universal set.

Properties

Let $A$ and $B$ be two sets in a universe $U$ . The following identities capture important properties of absolute complements:

De Morgan's laws:^[5]

$\left(A\cup B\right)^{\complement }=A^{\complement }\cap B^{\complement }.$
$\left(A\cap B\right)^{\complement }=A^{\complement }\cup B^{\complement }.$

Complement laws:^[5]

$A\cup A^{\complement }=U.$
$A\cap A^{\complement }=\emptyset .$
$\emptyset ^{\complement }=U.$
$U^{\complement }=\emptyset .$
${\text{If }}A\subseteq B{\text{, then }}B^{\complement }\subseteq A^{\complement }.$
(this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

$\left(A^{\complement }\right)^{\complement }=A.$

Relationships between relative and absolute complements:

$A\setminus B=A\cap B^{\complement }.$
$(A\setminus B)^{\complement }=A^{\complement }\cup B=A^{\complement }\cup (B\cap A).$

Relationship with a set difference:

$A^{\complement }\setminus B^{\complement }=B\setminus A.$

The first two complement laws above show that if $A$ is a non-empty, proper subset of $U$ , then ${A, A ∁}$ is a partition of $U$ .