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In
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concer ...
, the complement of a set , often denoted by (or ), is the set of
elements Element or elements may refer to: Science * Chemical element, a pure substance of one type of atom * Heating element, a device that generates heat by electrical resistance * Orbital elements, parameters required to identify a specific orbit of ...
not in . When all sets in the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. A ...
, i.e. all sets under consideration, are considered to be
members Member may refer to: * Military jury, referred to as "Members" in military jargon * Element (mathematics), an object that belongs to a mathematical set * In object-oriented programming, a member of a class ** Field (computer science), entries in ...
of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in .


Absolute complement


Definition

If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is usually denoted by Other notations include \overline A, A', \complement_U A, \text \complement A..


Examples

* Assume that the universe is the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. If is the set of odd numbers, then the complement of is the set of even numbers. If is the set of multiples of 3, then the complement of 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 is the suit of spades, then the complement of is the union of the suits of clubs, diamonds, and hearts. If the set is the union of the suits of clubs and diamonds, then the complement of is the union of the suits of hearts and spades.


Properties

Let and be two sets in a universe . The following identities capture important properties of absolute complements:
De Morgan's laws In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
: * \left(A \cup B \right)^\complement = A^\complement \cap B^\complement. * \left(A \cap B \right)^\complement = A^\complement \cup B^\complement. Complement laws: * A \cup A^\complement = U. * A \cap A^\complement = \varnothing . * \varnothing^\complement = U. * U^\complement = \varnothing. * \textA\subseteq B\textB^\complement \subseteq A^\complement. *: (this follows from the equivalence of a conditional with its contrapositive). Involution or double complement law: * \left(A^\complement\right)^\complement = A. Relationships between relative and absolute complements: * A \setminus B = A \cap B^\complement. * (A \setminus B)^\complement = A^\complement \cup B = A^\complement \cup (B \cap A). Relationship with a set difference: * A^\complement \setminus B^\complement = B \setminus A. The first two complement laws above show that if is a non-empty,
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of , then is a partition of .


Relative complement


Definition

If and are sets, then the relative complement of in ,. also termed the set difference of and ,. is the set of elements in but not in . The relative complement of in is denoted B \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 Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
) it can be interpreted as the set of all elements b - a, where is taken from and from . Formally: B \setminus A = \.


Examples

* \ \setminus \ = \. * \ \setminus \ = \ . * If \mathbb is the set of
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and \mathbb is the set of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s, then \mathbb\setminus\mathbb is the set of
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s.


Properties

Let , , and 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 = \empty. :* \empty \setminus A = \empty. :* A \setminus \empty = A. :* A \setminus U = \empty. :* If A\subset B, then C\setminus A\supset C\setminus B. :* A \supseteq B \setminus C is equivalent to C \supseteq B \setminus A.


Complementary relation

A
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
R is defined as a subset of a product of sets X \times Y. The complementary relation \bar is the set complement of R in X \times Y. The complement of relation R can be written \bar \ = \ (X \times Y) \setminus R. Here, R is often viewed as a
logical matrix A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1) matrix is a matrix with entries from the Boolean domain Such a matrix can be used to represent a binary relation between a pair of finite sets. Matrix representation ...
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 In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the r ...
are the elementary
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
s of the calculus of relations.


LaTeX notation

In the
LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosper ...
typesetting language, the command \setminus
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. A variant \smallsetminus is available in the amssymb package. The symbol \complement (as opposed to C) is produced by \complement. (It corresponds to the Unicode symbol ∁.)


In programming languages

Some
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming l ...
s have sets among their builtin
data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...
s. Such a data structure behaves as a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. ...
, 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
multiset In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that ...
s 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 do not have any data structure for sets.


See also

* * * * * *


Notes


References

* * *


External links

* * {{DEFAULTSORT:Complement (set theory) Basic concepts in set theory Operations on sets