Complement (set theory)

{{Short description|Set of the elements not in a given subset}} {{multiple image | align = right | image1 = Venn01.svg | width1 = 150 | alt1 = 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. | caption1 = If {{mvar|A}} is the area colored red in this image… | image2 = Venn10.svg | width2 = 150 | alt2 = 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. | caption2 = … then the complement of {{mvar|A}} is everything else. }}

In set theory, the '''complement''' of a set {{mvar|A}}, often denoted by <math>A^c</math> (or {{math|''A''′}}),<ref>{{Cite web|title=Complement and Set Difference|url=http://web.mnstate.edu/peil/MDEV102/U1/S6/Complement3.htm|access-date=2020-09-04|website=web.mnstate.edu}}</ref> is the set of elements not in {{mvar|A}}.<ref name=":1">{{Cite web|title=Complement (set) Definition (Illustrated Mathematics Dictionary)|url=https://www.mathsisfun.com/definitions/complement-set-.html|access-date=2020-09-04|website=www.mathsisfun.com}}</ref>

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

The '''relative complement''' of {{mvar|A}} with respect to a set {{mvar|B}}, also termed the '''set difference''' of {{mvar|B}} and {{mvar|A}}, written <math>B \setminus A,</math> is the set of elements in {{mvar|B}} that are not in {{mvar|A}}.

== Absolute complement ==  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}}:{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}}{{efn|1=The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}}}} <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>.{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}} Other notations include <math>\overline A </math>,{{sfn|Stoll|1979|p=[https://archive.org/details/settheorylogiced00robe/page/19 19]}} <math> 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 integers. 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 multiples of 3, then the complement of {{mvar|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 {{mvar|A}} is the suit of spades, then the complement of {{mvar|A}} is the 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 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:{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}} * <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:{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}} * <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 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 {{math|''U''}}.

== Relative complement ==

=== Definition === If {{math|''A''}} and {{math|''B''}} are sets, then the '''relative complement''' of {{math|''A''}} in {{math|''B''}},{{sfn|Halmos|1960|p=[https://books.google.com/books?id=lgdJDgAAQBAJ&pg=PA17 17]}} also termed the '''set difference''' of {{math|''B''}} and {{math|''A''}},{{sfn|Devlin|1979|p=6}} is the set of elements in {{math|''B''}} but not in {{math|''A''}}. thumb|230x230px|The '''relative complement''' of {{math|''A''}} in {{math|''B''}}: <math>B \cap A^c = B \setminus A</math> The relative complement of {{math|''A''}} in {{math|''B''}} is denoted <math>B \setminus A</math> according to the ISO 31-11 standard. It is sometimes written <math>B - A,</math> but this notation can be ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements <math>b - a,</math> where {{math|''b''}} is taken from {{math|''B''}} and {{math|''a''}} from {{math|''A''}}.

Formally: <math display=block>B \setminus A = \{ x\in B : x \notin A \}.</math>

=== Examples === * <math>\{ 1, 2, 3\} \setminus \{ 2,3,4\} = \{ 1 \}.</math> * <math>\{ 2, 3, 4 \} \setminus \{ 1,2,3 \} = \{ 4 \} .</math> * If <math>\mathbb{R}</math> is the set of real numbers and <math>\mathbb{Q}</math> is the set of rational numbers, then <math>\mathbb{R}\setminus\mathbb{Q}</math> is the set of irrational numbers.

=== Properties === {{See also|List of set identities and relations|Algebra of sets}}

Let {{math|''A''}}, {{math|''B''}}, and {{math|''C''}} be three sets in a universe {{mvar|U}}. The following identities capture notable properties of relative complements:

:* <math>C \setminus (A \cap B) = (C \setminus A) \cup (C \setminus B).</math> :* <math>C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B).</math> :* <math>C \setminus (B \setminus A) = (C \cap A) \cup (C \setminus B),</math> :*:with the important special case <math>C \setminus (C \setminus A) = (C \cap A)</math> demonstrating that intersection can be expressed using only the relative complement operation. :* <math>(B \setminus A) \cap C = (B \cap C) \setminus A = B \cap (C \setminus A).</math> :* <math>(B \setminus A) \cup C = (B \cup C) \setminus (A \setminus C).</math> :* <math>A \setminus A = \emptyset.</math> :* <math>\empty \setminus A = \empty.</math> :* <math>A \setminus \empty = A.</math> :* <math>A \setminus U = \empty.</math> :* If <math>A\subset B</math>, then <math>C\setminus A\supset C\setminus B</math>. :* <math>A \supseteq B \setminus C</math> is equivalent to <math>C \supseteq B \setminus A</math>.

== Complementary relation == A binary relation <math>R</math> is defined as a subset of a product of sets <math>X \times Y.</math> The '''complementary relation''' <math>\bar{R}</math> is the set complement of <math>R</math> in <math>X \times Y.</math> The complement of relation <math>R</math> can be written <math display=block>\bar{R} \ = \ (X \times Y) \setminus R.</math> Here, <math>R</math> is often viewed as a logical matrix with rows representing the elements of <math>X,</math> and columns elements of <math>Y.</math> The truth of <math>aRb</math> corresponds to 1 in row <math>a,</math> column <math>b.</math> Producing the complementary relation to <math>R</math> then corresponds to switching all 1s to 0s, and 0s to 1s for the logical matrix of the complement.

Together with composition of relations and converse relations, complementary relations and the algebra of sets are the elementary operations of the calculus of relations.

== LaTeX notation == {{See also|List of mathematical symbols by subject}}

In the LaTeX typesetting language, the command <code>\setminus</code><ref name="The Comprehensive LaTeX Symbol List">[http://ctan.unsw.edu.au/info/symbols/comprehensive/symbols-a4.pdf] {{Webarchive|url=https://web.archive.org/web/20220305100117/http://ctan.unsw.edu.au/info/symbols/comprehensive/symbols-a4.pdf |date=2022-03-05 }} The Comprehensive LaTeX Symbol List</ref> is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the <code>\setminus</code> command looks identical to <code>\backslash</code>, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence <code>\mathbin{\backslash}</code>. A variant <code>\smallsetminus</code> is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol <math>\complement</math> (as opposed to <math>C</math>) is produced by <code>\complement</code>. (It corresponds to the Unicode symbol {{unichar|2201|COMPLEMENT}}.)

== See also ==

* {{annotated link|Algebra of sets}} * {{annotated link|Intersection (set theory)}} * {{annotated link|List of set identities and relations}} * {{annotated link|Naive set theory}} * {{annotated link|Symmetric difference}} * {{annotated link|Union (set theory)}}

== Footnotes == {{notelist|group=lower-alpha}}

== Notes == {{reflist}}

== References ==

* {{cite book | last = Bourbaki | first = N. | author-link = Nicolas Bourbaki | title = Théorie des ensembles | publisher = Hermann | place = Paris | year = 1970 | isbn = 978-3-540-34034-8 | language = fr }} * {{cite book | last = Devlin | first = Keith J. | author-link = Keith Devlin | title = Fundamentals of contemporary set theory | series = Universitext | publisher = Springer | year = 1979 | isbn = 0-387-90441-7 | zbl = 0407.04003 }} * {{cite book | last = Halmos | first = Paul R. | author-link = Paul Halmos | title = Naive set theory | url = https://archive.org/details/naivesettheory0000halm | url-access = registration | series = The University Series in Undergraduate Mathematics | publisher = van Nostrand Company | year = 1960 | isbn = 9780442030643 | zbl = 0087.04403 }} * {{cite book | last = Stoll | first = Robert R. | title = Set Theory and Logic | location = Mineola, N.Y. | publisher = Dover Publications | year = 1979 | isbn = 0-486-63829-4 | url = https://archive.org/details/settheorylogiced00robe }}

==External links== * {{MathWorld |title=Complement |id=Complement }} * {{MathWorld |title=Complement Set |id=ComplementSet }}

Category:Basic concepts in set theory Category:Operations on sets