Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complement of A with respect to a set B 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 all elements in U but not in A.

Relative complement

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^c = B \setminus A

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

Formally:

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

Examples:

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

The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.

If A, B, and C are sets, then the following identities hold:

C ∖ (A ∩ B) = (C ∖ A) ∪ (C ∖ B),
C ∖ (A ∪ B) = (C ∖ A) ∩ (C ∖ B),
C ∖ (B ∖ A) = (C ∩ A) ∪ (C ∖ B),
with the important special case C ∖ (C ∖ A) = C ∩ A demonstrating that intersection can be expressed using only the relative complement operation,
(B ∖ A) ∩ C = (B ∩ C) ∖ A = B∩(C ∖ A),
(B ∖ A) ∪ C = (B ∪ C) ∖ (A ∖ C),
A ∖ A = ∅,
∅ ∖ A = ∅,
A ∖ ∅ = A.

Absolute complement

The absolute complement of

A

in U:

Ac = U \ A

If a universe $U$ is defined, then the relative complement of $A$ in $U$ is called the absolute complement (or simply complement) of $A$ , and is denoted by $A c$ or sometimes $A'$ . The same set often^[3] is denoted by $\complement_U A$ or $\complement A$ if $U$ is fixed, that is:

A c = U ∖ A

For example, if the universe is the set of integers, then the complement of the set of odd numbers is the set of even numbers.

The following lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.

If $A$ and $B$ are subsets of a universe $U$ , then the following identities hold:

De Morgan's laws:^[1]

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

Complement laws:^[1]

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

Involution or double complement law:

$\left(A^{c}\right)^{c}=A.$

Relationships between relative and absolute complements:

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

Relationship with set difference:

$A^c \setminus B^c = B \setminus A.$

The first two complement laws above shows that if $A$ is a non-empty, proper subset of $U$ , then ${A, A c$ } is a partition of $U$ .

Notation

In the LaTeX typesetting language, the command \setminus^[4] 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)^[5]

Common Lisp: set-difference, nset-difference^[6]

F#: Set.difference a b^[7]

a - b^[8]

Falcon: diff = a - b^[9]

Haskell: difference a b; a \\ b^[10]

Java: diff = a.clone();; diff.removeAll(b);^[11]

Julia: setdiff^[12]

Mathematica: Complement^[13]

MATLAB: setdiff^[14]

OCaml: Set.S.diff^[15]

Octave: setdiff^[16]

PARI/GP: setminus^[17]

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);^[18]

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

Python: diff = a.difference(b)^[19]; diff = a - b^[19]

R: setdiff^[20]

Racket: (set-subtract a b)^[21]

Ruby: diff = a - b^[22]

Scala: a.diff(b)^[23]

a -- b^[23]

Smalltalk (Pharo): a difference: b

SQL: SELECT * FROM A



EXCEPT
SELECT * FROM B

Unix shell: comm -23 a b^[24]; grep -vf b a # less efficient, but works with small unsorted sets

References

1 2 3 Halmos (1960) p.17
↑ Devlin (1979) p. 6.
↑ Bourbaki p. E II.6
↑ The Comprehensive LaTeX Symbol List
↑ 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.
↑ . 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).
↑ . GNU Octave Reference Manual
↑ PARI/GP User's Manual
↑ PHP: array_diff, PHP Manual
1 2 . Python v2.7.3 documentation. Accessed on January 17, 2013.
↑ R Reference manual p. 410.
↑ . The Racket Reference. Accessed on May 19, 2015.
↑ Class: Array Ruby Documentation
1 2 scala.collection.Set. Scala Standard Library 2.11.7, Accessed on July 12, 2015.
↑ comm(1), Unix Seventh Edition Manual, 1979.

Halmos, Paul R. (1960). Naive set theory. The University Series in Undergraduate Mathematics. van Nostrand Company. Zbl 0087.04403.
Devlin, Keith J. (1979). Fundamentals of contemporary set theory. Universitext. Springer-Verlag. ISBN 0-387-90441-7. Zbl 0407.04003.
Bourbaki, N. (1970). Théorie des ensembles (in French). Paris: Hermann. ISBN 978-3-540-34034-8.

External links

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

Set theory

Axioms	Choice countable dependent 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 Lotfi A. Zadeh Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Willard Quine

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