Complement (set theory)
In set theory, the complement of a set A refers to elements 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.
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.
Contents
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. 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:^{[1]}
 .
Formally:
The absolute complement of A is usually denoted by . Other notations include , , , , and .^{[2]}
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 52card 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
Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:
 De Morgan's laws:^{[3]}
 Complement laws:^{[3]}



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

 Involution or double complement law:
 Relationships between relative and absolute complements:
 Relationship with set difference:

The first two complement laws above show that if A is a nonempty, proper subset of U, then {A, A^{∁}} is a partition of U.
Relative complement
Definition
If A and B are sets, then the relative complement of A in B,^{[3]} also termed the settheoretic difference of B and A,^{[4]} is the set of elements in B but not in A.
The relative complement of A in B is denoted B ∖ A according to the ISO 3111 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:
Examples
 .
 .
 If is the set of real numbers and is the set of rational numbers, then is the set of irrational numbers.
Properties
Let A, B, and C be three sets. The following identities capture notable properties of relative complements:

 .
 .

,
 with the important special case demonstrating that intersection can be expressed using only the relative complement operation.
 .
 .
 .
 .
 .
 .
LaTeX notation
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
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
a.Except(b);
 C++
set_difference(a.begin(), a.end(), b.begin(), b.end(), result.begin());
 Clojure

(clojure.set/difference a b)
^{[6]}
 Common Lisp

setdifference, nsetdifference
^{[7]}
 F#

Set.difference a b
^{[8]} 
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

(setsubtract a b)
^{[22]}
 Ruby

diff = a  b
^{[23]}
 Rust

let diff = std::collections::HashSet::difference(&hashset_a, &hashset_b);
^{[24]}
 Scala

a.diff(b)
^{[25]} 
a  b
^{[25]}
 SQL

SELECT * FROM A EXCEPT SELECT * FROM B
 Unix shell

comm 23 a b
^{[26]} grep vFxf b a # less efficient, but works with small unsorted sets
 XQuery (and XPath starting from version 2.0)

$a except $b
^{[27]}
 Z Shell

diff=("${(@)a:b}")
^{[28]}
See also
Notes
 ^ 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.
 ^ ^{a} ^{b} ^{c} Halmos 1960, p. 17.
 ^ Devlin 1979, p. 6.
 ^ [1] The Comprehensive LaTeX Symbol List
 ^ [2] clojure.set API reference
 ^ Common Lisp HyperSpec, Function setdifference, nsetdifference. Accessed on September 8, 2009.
 ^ Set.difference Function (F#). Accessed on July 12, 2015.
 ^ Set.(  ) Method (F#). Accessed on July 12, 2015.
 ^ Array subtraction, data structures. Accessed on July 28, 2014.
 ^ Data.Set (Haskell) Archived 20090528 at the Wayback Machine.
 ^ 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 LibraryJulia 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
 ^ [7] The Rust Standard Library Documentation. Accessed on March 7, 2017.
 ^ ^{a} ^{b} scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.
 ^ comm(1) Archived 20081019 at the Wayback Machine., Unix Seventh Edition Manual, 1979.
 ^ "XQuery 1.0 and XPath 2.0 Functions and Operators (Second Edition)". W3C. 13 April 2015.
 ^ Parameter Expansion, The Z Shell Manual.
References
 Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 9783540340348.
 Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer. ISBN 0387904417. Zbl 0407.04003.
 Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.
External links
 Weisstein, Eric W. "Complement". MathWorld.
 Weisstein, Eric W. "Complement Set". MathWorld.