complement (set theory)#Complementary relation

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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}}:The set in which the complement is considered is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.

$$A^c=\; U\; \backslash setminus\; A\; =\; \backslash \{\; x\; \backslash in\; U\; :\; x\; \backslash notin\; A\; \backslash \}.$$

The absolute complement of {{mvar|A}} is usually denoted by $A^c$. Other notations include $\backslash overline\; A,\; A\text{'},$ $\backslash complement\_U\; A,\; \backslash text\{\; and\; \}\; \backslash complement\; A.${{harvnb|Bourbaki|1970|p=E II.6}}.

• 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.

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:

• $\backslash left(A\; \backslash cup\; B\; \backslash right)^c=\; A^c\; \backslash cap\; B^c.$
• $\backslash left(A\; \backslash cap\; B\; \backslash right)^c\; =\; A^c\; \backslash cup\; B^c.$

Complement laws:

• $A\; \backslash cup\; A^c\; =\; U.$
• $A\; \backslash cap\; A^c\; =\; \backslash empty\; .$
• $\backslash empty^c\; =\; U.$
• $U^c\; =\; \backslash empty.$
• $\backslash text\{If\; \}A\backslash subseteq\; B\backslash text\{,\; then\; \}B^c\; \backslash subseteq\; A^c.$
• : (this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

• $\backslash left(A^c\backslash right)^c\; =\; A.$

Relationships between relative and absolute complements:

• $A\; \backslash setminus\; B\; =\; A\; \backslash cap\; B^c.$
• $(A\; \backslash setminus\; B)^c\; =\; A^c\; \backslash cup\; B\; =\; A^c\; \backslash cup\; (B\; \backslash cap\; A).$

Relationship with a set difference:

• $A^c\; \backslash setminus\; B^c\; =\; B\; \backslash setminus\; A.$

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

If {{math|A}} and {{math|B}} are sets, then the relative complement of {{math|A}} in {{math|B}},{{harvnb|Halmos|1960|p=17}}. also termed the set difference of {{math|B}} and {{math|A}},{{harvnb|Devlin|1979|p=6}}. is the set of elements in {{math|B}} but not in {{math|A}}.

File:Relative compliment.svg

The relative complement of {{math|A}} in {{math|B}} is denoted $B\; \backslash setminus\; A$ according to the ISO 31-11 standard. It is sometimes written $B\; -\; A,$ but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements $b\; -\; a,$ where {{math|b}} is taken from {{math|B}} and {{math|a}} from {{math|A}}.

Formally:

$$B\; \backslash setminus\; A\; =\; \backslash \{\; x\backslash in\; B\; :\; x\; \backslash notin\; A\; \backslash \}.$$

• $\backslash \{\; 1,\; 2,\; 3\backslash \}\; \backslash setminus\; \backslash \{\; 2,3,4\backslash \}\; =\; \backslash \{\; 1\; \backslash \}.$
• $\backslash \{\; 2,\; 3,\; 4\; \backslash \}\; \backslash setminus\; \backslash \{\; 1,2,3\; \backslash \}\; =\; \backslash \{\; 4\; \backslash \}\; .$
• If $\backslash mathbb\{R\}$ is the set of real numbers and $\backslash mathbb\{Q\}$ is the set of rational numbers, then $\backslash mathbb\{R\}\backslash setminus\backslash mathbb\{Q\}$ is the set of irrational numbers.

{{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:

:* $C\; \backslash setminus\; (A\; \backslash cap\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cup\; (C\; \backslash setminus\; B).$

:* $C\; \backslash setminus\; (A\; \backslash cup\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cap\; (C\; \backslash setminus\; B).$

:* $C\; \backslash setminus\; (B\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)\; \backslash cup\; (C\; \backslash setminus\; B),$

:*:with the important special case $C\; \backslash setminus\; (C\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)$ demonstrating that intersection can be expressed using only the relative complement operation.

:* $(B\; \backslash setminus\; A)\; \backslash cap\; C\; =\; (B\; \backslash cap\; C)\; \backslash setminus\; A\; =\; B\; \backslash cap\; (C\; \backslash setminus\; A).$

:* $(B\; \backslash setminus\; A)\; \backslash cup\; C\; =\; (B\; \backslash cup\; C)\; \backslash setminus\; (A\; \backslash setminus\; C).$

:* $A\; \backslash setminus\; A\; =\; \backslash emptyset.$

:* $\backslash empty\; \backslash setminus\; A\; =\; \backslash empty.$

:* $A\; \backslash setminus\; \backslash empty\; =\; A.$

:* $A\; \backslash setminus\; U\; =\; \backslash empty.$

:* If $A\backslash subset\; B$, then $C\backslash setminus\; A\backslash supset\; C\backslash setminus\; B$.

:* $A\; \backslash supseteq\; B\; \backslash setminus\; C$ is equivalent to $C\; \backslash supseteq\; B\; \backslash setminus\; A$.

A binary relation $R$ is defined as a subset of a product of sets $X\; \backslash times\; Y.$ The complementary relation $\backslash bar\{R\}$ is the set complement of $R$ in $X\; \backslash times\; Y.$ The complement of relation $R$ can be written

$$\backslash bar\{R\}\; \backslash \; =\; \backslash \; (X\; \backslash times\; Y)\; \backslash setminus\; R.$$

Here, $R$ is often viewed as a logical matrix with rows representing the elements of $X,$ and columns elements of $Y.$ The truth of $aRb$ corresponds to 1 in row $a,$ column $b.$ Producing the complementary relation to $R$ 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.

{{See also|List of mathematical symbols by subject}}

In the LaTeX typesetting language, the command \setminus[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 is usually used for rendering a set difference symbol, which is similar to a backslash symbol. When rendered, the \setminus command looks identical to \backslash, except that it has a little more space in front and behind the slash, akin to the LaTeX sequence \mathbin{\backslash}. A variant \smallsetminus is available in the amssymb package, but this symbol is not included separately in Unicode. The symbol $\backslash complement$ (as opposed to $C$) is produced by \complement. (It corresponds to the Unicode symbol {{unichar|2201|COMPLEMENT}}.)

• {{annotated link|Algebra of sets}}
• {{annotated link|Intersection (set theory)}}
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• {{annotated link|Naive set theory}}
• {{annotated link|Symmetric difference}}
• {{annotated link|Union (set theory)}}

{{reflist}}

{{reflist|group=note}}

• {{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

}}

• {{MathWorld |title=Complement |id=Complement }}
• {{MathWorld |title=Complement Set |id=ComplementSet }}

{{Set theory}}

{{Mathematical logic}}

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Category:Basic concepts in set theory

Category:Operations on sets