Editing Complement (set theory) (section)

== Absolute complement ==
<!-- This section is linked from [[Bayes' theorem]] and [[absolute set complement]] -->
[[File:Venn10.svg|150px|thumb|The '''absolute complement''' of the white disc is the red region]]

=== Definition ===
If {{mvar|A}} is a set, then the '''absolute complement''' of {{mvar|A}} (or simply the '''complement''' of {{mvar|A}}) is the set of elements not in {{mvar|A}} (within a larger set that is implicitly defined). In other words, let {{mvar|U}} be a set that contains all the elements under study; if there is no need to mention {{mvar|U}}, either because it has been previously specified, or it is obvious and unique, then the absolute complement of {{mvar|A}} is the relative complement of {{mvar|A}} in {{mvar|U}}:<ref>The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.</ref>
<math display=block>A^c= U \setminus A = \{ x \in U : x \notin A \}.</math>

The absolute complement of {{mvar|A}} is usually denoted by <math>A^c</math>. Other notations include <math>\overline A, A',</math><ref name=":1" /> <math>\complement_U A, \text{ and } \complement A.</math><ref name="Bou">{{harvnb|Bourbaki|1970|p=E II.6}}.</ref>

=== Examples ===
* Assume that the universe is the set of [[integer]]s. If {{mvar|A}} is the set of odd numbers, then the complement of {{mvar|A}} is the set of even numbers. If {{mvar|B}} is the set of [[Multiple (mathematics)|multiples]] of 3, then the complement of {{mvar|B}} is the set of numbers [[Modular arithmetic|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 {{mvar|A}} is the suit of spades, then the complement of {{mvar|A}} is the [[Union (set theory)|union]] of the suits of clubs, diamonds, and hearts. If the set {{mvar|B}} is the union of the suits of clubs and diamonds, then the complement of {{mvar|B}} is the union of the suits of hearts and spades.
*When the universe is the [[Universe (mathematics)|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 {{mvar|A}} and {{mvar|B}} be two sets in a universe {{mvar|U}}. The following identities capture important properties of absolute complements:

[[De Morgan's laws]]:<ref name="Halmos-1960" />
* <math>\left(A \cup B \right)^c= A^c \cap B^c.</math>
* <math>\left(A \cap B \right)^c = A^c \cup B^c.</math>

Complement laws:<ref name="Halmos-1960" />
* <math>A \cup A^c = U.</math>
* <math>A \cap A^c = \empty .</math>
* <math>\empty^c = U.</math>
* <math> U^c = \empty.</math>
* <math>\text{If }A\subseteq B\text{, then }B^c \subseteq A^c.</math>
*: (this follows from the equivalence of a conditional with its [[contrapositive]]).

[[Involution (mathematics)|Involution]] or double complement law:
* <math>\left(A^c\right)^c = A.</math>

Relationships between relative and absolute complements:
* <math>A \setminus B = A \cap B^c.</math>
* <math>(A \setminus B)^c = A^c \cup B = A^c \cup (B \cap A).</math>

Relationship with a set difference:
* <math> A^c \setminus B^c = B \setminus A. </math>

The first two complement laws above show that if {{math|''A''}} is a non-empty, [[proper subset]] of {{math|''U''}}, then {{math|{''A'', ''A''<sup>∁</sup>}{{null}}}} is a [[Partition of a set|partition]] of {{math|''U''}}.