Complement (set theory)

In set theory, the complement of a set $A$ refers to elements not in $A$ . The relative complement of $A$ with respect to a set $B$ , also termed the difference of sets $A$ and $B$ , written $B ∖ A$ , is the set of elements in $B$ but not in $A$ . When all sets under consideration are considered to be subsets of a given set $U$ , the absolute complement of $A$ is the set of elements in $U$ but not in $A$ .

Relative complement[edit]

Definition[edit]

If $A$ and $B$ are sets, then the relative complement of $A$ in $B$ ,^[1] also termed the set-theoretic difference of $B$ and $A$ ,^[2] is the set of elements in $B$ but not in $A$ .

The relative complement of

A

(left circle) in

B

(right circle):

B\cap A^{\complement }=B\setminus A

The relative complement of $A$ in $B$ is denoted $B ∖ A$ according to the ISO 31-11 standard. It is sometimes written $B - A$ , but this notation is ambiguous, as in some contexts it can be interpreted as the set of all elements $b - a$ , where $b$ is taken from $B$ and $a$ from $A$ .

Formally:

B\setminus A=\{x\in B\mid x\notin A\}.

Examples[edit]

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

Properties[edit]

Let $A$ , $B$ , and $C$ be three sets. The following identities capture notable properties of relative complements:

$C\setminus (A\cap B)=(C\setminus A)\cup (C\setminus B)$ .
$C\setminus (A\cup B)=(C\setminus A)\cap (C\setminus B)$ .
$C\setminus (B\setminus A)=(C\cap A)\cup (C\setminus B)$ ,

with the important special case $C\setminus (C\setminus A)=(C\cap A)$ demonstrating that intersection can be expressed using only the relative complement operation.
$(B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A)$ .
$(B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C)$ .
$A\setminus A=\emptyset$ .
$\emptyset \setminus A=\emptyset$ .
$A\setminus \emptyset =A$ .

Absolute complement[edit]

The absolute complement of

A

in U:

A^{\complement }=U\setminus A

Definition[edit]

If $A$ is a set, then the absolute complement of $A$ (or simply the complement of $A$ ) is the set of elements not in $A$ . In other words, if $U$ is the universe that contains all the elements under study, and there is no need to mention it because 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

.

Formally:

A^{\complement }=\{x\in U\mid x\notin A\}.

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

Examples[edit]

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.
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.

Properties[edit]

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

De Morgan's laws:^[1]

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

Complement laws:^[1]

$A\cup A^{\complement }=U.$
$A\cap A^{\complement }=\varnothing .$
$\varnothing ^{\complement }=U.$
$U^{\complement }=\varnothing .$
${\text{If }}A\subset B{\text{, then }}B^{\complement }\subset A^{\complement }.$

(this follows from the equivalence of a conditional with its contrapositive).

Involution or double complement law:

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

Relationships between relative and absolute complements:

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

Relationship with 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 $U$ .

LaTeX notation[edit]

In the LaTeX typesetting language, the command \setminus^[5] 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.

Complements in various programming languages[edit]

Some programming languages allow for manipulation of sets as data structures, using these operators or functions to construct the difference of sets a and b:

.NET Framework: b.Except(a);

C++: set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());

Clojure: (clojure.set/difference a b)^[6]

Common Lisp: set-difference, nset-difference^[7]

F#: Set.difference a b^[8]

or

a - b^[9]

Falcon: diff = a - b^[10]

Haskell: difference a b; a \\ b^[11]

Java: diff = a.clone();; diff.removeAll(b);^[12]

Julia: setdiff^[13]

Mathematica: Complement^[14]

MATLAB: setdiff^[15]

OCaml: Set.S.diff^[16]

Octave: setdiff^[17]

PARI/GP: setminus^[18]

Pascal: SetDifference := a - b;

Perl 5

# for perl version >= 5.10
@a = grep {not $_ ~~ @b} @a;

Perl 6

$A ∖ $B
$A (-) $B # texas version

PHP: array_diff($a, $b);^[19]

Prolog: a(X),\+ b(X).

Python: diff = a.difference(b)^[20]; diff = a - b^[20]

R: setdiff^[21]

Racket: (set-subtract a b)^[22]

Ruby: diff = a - b^[23]

Scala: a.diff(b)^[24]

or

a -- b^[24]

Smalltalk (Pharo): a difference: b

SQL

SELECT * FROM A
EXCEPT
SELECT * FROM B

Unix shell: comm -23 a b^[25]; grep -vf b a # less efficient, but works with small unsorted sets

Notes[edit]

^ ^a ^b ^c Halmos 1960, p. 17.
^ Devlin 1979, p. 6.
^ The set other than $A$ is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.
^ Bourbaki 1970, p. E II.6.
^ [1] The Comprehensive LaTeX Symbol List
^ [2] clojure.set API reference
^ Common Lisp HyperSpec, Function set-difference, nset-difference. Accessed on September 8, 2009.
^ Set.difference<'T> Function (F#). Accessed on July 12, 2015.
^ Set.( - )<'T> Method (F#). Accessed on July 12, 2015.
^ Array subtraction, data structures. Accessed on July 28, 2014.
^ Data.Set (Haskell)
^ Set (Java 2 Platform SE 5.0). JavaTM 2 Platform Standard Edition 5.0 API Specification, updated in 2004. Accessed on February 13, 2008.
^ [3]. The Standard Library--Julia Language documentation. Accessed on September 24, 2014
^ Complement. Mathematica Documentation Center for version 6.0, updated in 2008. Accessed on March 7, 2008.
^ Setdiff. MATLAB Function Reference for version 7.6, updated in 2008. Accessed on May 19, 2008.
^ Set.S (OCaml).
^ [4]. GNU Octave Reference Manual
^ PARI/GP User's Manual Archived September 11, 2015, at the Wayback Machine.
^ PHP: array_diff, PHP Manual
^ ^a ^b [5]. Python v2.7.3 documentation. Accessed on January 17, 2013.
^ R Reference manual p. 410.
^ [6]. The Racket Reference. Accessed on May 19, 2015.
^ Class: Array Ruby Documentation
^ ^a ^b scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.
^ comm(1), Unix Seventh Edition Manual, 1979.

References[edit]

Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.

Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer. ISBN 0-387-90441-7. Zbl 0407.04003.

Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.

External links[edit]

Weisstein, Eric W. "Complement". MathWorld.

Weisstein, Eric W. "Complement Set". MathWorld.

[Halmos-1960-1] Halmos 1960, p. 17.

[2] Devlin 1979, p. 6.

[3] The set other than $A$ is thus implicitly mentioned in an absolute complement, and explicitly mentioned in a relative complement.

[Bou-4] Bourbaki 1970, p. E II.6.

[The_Comprehensive_LaTeX_Symbol_List-5] [1] The Comprehensive LaTeX Symbol List

[6] [2] clojure.set API reference

[CLHS_set-difference-7] Common Lisp HyperSpec, Function set-difference, nset-difference. Accessed on September 8, 2009.

[FSharp_Set-difference-8] Set.difference<'T> Function (F#). Accessed on July 12, 2015.

[FSharp_Set-minus-operator-9] Set.( - )<'T> Method (F#). Accessed on July 12, 2015.

[Falcon_Array_Subtraction-10] Array subtraction, data structures. Accessed on July 28, 2014.

[Data.Set-11] Data.Set (Haskell)

[J2SE_Set-12] Set (Java 2 Platform SE 5.0). JavaTM 2 Platform Standard Edition 5.0 API Specification, updated in 2004. Accessed on February 13, 2008.

[Julia_set-13] [3]. The Standard Library--Julia Language documentation. Accessed on September 24, 2014

[Mathematica_set-14] Complement. Mathematica Documentation Center for version 6.0, updated in 2008. Accessed on March 7, 2008.

[MATLAB_set-15] Setdiff. MATLAB Function Reference for version 7.6, updated in 2008. Accessed on May 19, 2008.

[OCaml_Set_S_diff-16] Set.S (OCaml).

[17] [4]. GNU Octave Reference Manual

[18] PARI/GP User's Manual Archived September 11, 2015, at the Wayback Machine.

[19] PHP: array_diff, PHP Manual

[Python_set-20] [5]. Python v2.7.3 documentation. Accessed on January 17, 2013.

[21] R Reference manual p. 410.

[22] [6]. The Racket Reference. Accessed on May 19, 2015.

[23] Class: Array Ruby Documentation

[Scala_Set-24] scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.

[Unix_comm-25] (1), Unix Seventh Edition Manual, 1979.

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v t e Set theory

Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification

Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union

Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram

Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal

Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck

Paradoxes Problems	Russell's paradox Suslin's problem

Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

Complement (set theory)

Contents

Relative complement[edit]

Definition[edit]

Examples[edit]

Properties[edit]

Absolute complement[edit]

Definition[edit]

Examples[edit]

Properties[edit]

LaTeX notation[edit]

Complements in various programming languages[edit]

See also[edit]

Notes[edit]

References[edit]

External links[edit]

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