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In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), 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, i ...
, the complement of a set , often denoted by (or ), are the elements not in . When all sets under consideration are considered to be
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
s 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^c = U \setminus A. Or formally: A^c = \. 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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
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 The standard 52-card deck of French-suited playing cards French-suited playing cards or French-suited cards are playing cards, cards that use the French Suit (cards), suits of (clovers or clubs ), (tiles or diamonds ), (hearts ) ...
. 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 calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...
: * \left(A \cup B \right)^c=A^c \cap B^c. * \left(A \cap B \right)^c=A^c\cup B^c. Complement laws: * A \cup A^c = U . * A \cap A^c =\varnothing . * \varnothing^c =U. * U^c =\varnothing. * \textA\subseteq B\textB^c\subseteq A^c. *: (this follows from the equivalence of a conditional with its
contrapositive In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
).
Involution Involution may refer to: * Involute, a construction in the differential geometry of curves * ''Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour input ...
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 = A^c \cup (B \cap A). Relationship with a set difference: * A^c \setminus B^c = B \setminus A. The first two complement laws above show that if is a non-empty,
proper subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of , then is a 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 . 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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), ...
) 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s, then \mathbb\setminus\mathbb is the set of
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
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 matrixA logical matrix, binary matrix, relation matrix, Boolean matrix, or (0,1) matrix is a matrix with entries from the Boolean domain In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number ...
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 In the of s, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the , the composition of relations is called relative multiplication, and its result is called a relative produc ...
and
converse relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, complementary relations and the
algebra of sets In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
are the elementary operations of the
calculus of relations In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
.


LaTeX notation

In the
LaTeX Latex is a stable dispersion (emulsion An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class o ...

LaTeX
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 The backslash is a typographical mark used mainly in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and develop ...
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.


In programming languages

Some
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

programming language
s have sets among their builtin
data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of ...

data structure
s. Such a data structure behaves as a
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 ARRAY, also known as ARRAY Now, is an independent distribution company launched by film maker and former publicist Ava DuVernay Ava Marie DuVernay (; born August 24, 1972) is an American filmmaker. She won the directing award in the U.S. drama ...
. 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) Operations on sets