Complement (set theory)

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If

A

is the area colored red in this image...

... then the complement of

A

is everything else.

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

1 Absolute complement
2 Relative complement
3 Complementary relation
4 LaTeX notation
5 In programming languages
6 See also
7 Notes
8 References
9 External links

Absolute complement[edit]

The absolute complement of

A

(left circle) in U:

A^{c}=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 $U$ :^[1]

A^{c}=U\setminus A

.

Formally:

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

The absolute complement of $A$ is usually denoted by $A^{c}$ . Other notations include ${\overline {A}}$ , $A'$ , $\complement _{U}A$ , and $\complement A$ .^[2]

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 (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 $A$ 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:^[3]

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

Complement laws:^[3]

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

Involution or double complement law:

$(A^{c})^{c}=A.$

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 show that if $A$ is a non-empty, proper subset of $U$ , then ${A, A ∁}$ is a partition of $U$ .

Relative complement[edit]

Definition[edit]

If $A$ and $B$ are sets, then the relative complement of $A$ in $B$ ,^[3] also termed the set difference of $B$ and $A$ ,^[4] 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. 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.
$(B\setminus A)\cap C=(B\cap C)\setminus A=B\cap (C\setminus A)$ .
$(B\setminus A)\cup C=(B\cup C)\setminus (A\setminus C)$ .
$A\setminus A=\emptyset$ .
$\emptyset \setminus A=\emptyset$ .
$A\setminus \emptyset =A$ .
$A\setminus U=\emptyset$ .

Complementary relation[edit]

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

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

Often R is 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.

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.

In programming languages[edit]

Some programming languages have sets among their builtin data structures. Such a data structure behaves as a finite set, that is it consists of a finite number of data, that are not specifically ordered, and may thus be considered as the elements of a set. In some cases, the elements are not necessary distinct, and the data structure codes multisets rather than sets. These programming languages have operators or functions for computing the complement and the set differences.

These operators may generally be applied also to data structures that are not really mathematical sets, such as ordered lists or arrays. It follows that some programming languages may have a function called "set_difference" even if they have not any data structure for sets.

See also[edit]

Notes[edit]

^ 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

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]

v t e Set theory
Axioms	Adjunction Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size 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 (Dedekind-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 Burali-Forti paradox
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

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