In

The Comprehensive LaTeX Symbol List is usually used for rendering a set difference symbol, which is similar to a

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 ...

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 $B\; \backslash setminus\; A,$ 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 : $$A^c\; =\; U\; \backslash setminus\; A.$$ Or formally: $$A^c\; =\; \backslash .$$ The absolute complement of is usually denoted by Other notations include $\backslash overline\; A,\; A\text{'},$ $\backslash complement\_U\; A,\; \backslash text\; \backslash complement\; A.$.Examples

* Assume that the universe is the set ofinteger
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 ...

:
* $\backslash left(A\; \backslash cup\; B\; \backslash right)^c=A^c\; \backslash cap\; B^c.$
* $\backslash left(A\; \backslash cap\; B\; \backslash right)^c=A^c\backslash cup\; B^c.$
Complement laws:
* $A\; \backslash cup\; A^c\; =\; U\; .$
* $A\; \backslash cap\; A^c\; =\backslash varnothing\; .$
* $\backslash varnothing^c\; =U.$
* $U^c\; =\backslash varnothing.$
* $\backslash textA\backslash subseteq\; B\backslash textB^c\backslash subseteq\; A^c.$
*: (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:
* $\backslash left(A^c\backslash right)^c\; =\; A.$
Relationships between relative and absolute complements:
* $A\; \backslash setminus\; B\; =\; A\; \backslash cap\; B^c.$
* $(A\; \backslash setminus\; B)^c\; =\; A^c\; \backslash cup\; B\; =\; A^c\; \backslash cup\; (B\; \backslash cap\; A).$
Relationship with a set difference:
* $A^c\; \backslash setminus\; B^c\; =\; B\; \backslash setminus\; A.$
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 $B\; \backslash setminus\; A$ according to the . It is sometimes written $B\; -\; A,$ but this notation is ambiguous, as in some contexts (for example, Minkowski set operations infunctional 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:
$$B\; \backslash setminus\; A\; =\; \backslash .$$
Examples

* $\backslash \; \backslash setminus\; \backslash \; =\; \backslash .$ * $\backslash \; \backslash setminus\; \backslash \; =\; \backslash \; .$ * If $\backslash mathbb$ is the set ofreal 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 $\backslash mathbb$ 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 $\backslash mathbb\backslash setminus\backslash mathbb$ 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: :* $C\; \backslash setminus\; (A\; \backslash cap\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cup\; (C\; \backslash setminus\; B).$ :* $C\; \backslash setminus\; (A\; \backslash cup\; B)\; =\; (C\; \backslash setminus\; A)\; \backslash cap\; (C\; \backslash setminus\; B).$ :* $C\; \backslash setminus\; (B\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)\; \backslash cup\; (C\; \backslash setminus\; B),$ :*:with the important special case $C\; \backslash setminus\; (C\; \backslash setminus\; A)\; =\; (C\; \backslash cap\; A)$ demonstrating that intersection can be expressed using only the relative complement operation. :* $(B\; \backslash setminus\; A)\; \backslash cap\; C\; =\; (B\; \backslash cap\; C)\; \backslash setminus\; A\; =\; B\; \backslash cap\; (C\; \backslash setminus\; A).$ :* $(B\; \backslash setminus\; A)\; \backslash cup\; C\; =\; (B\; \backslash cup\; C)\; \backslash setminus\; (A\; \backslash setminus\; C).$ :* $A\; \backslash setminus\; A\; =\; \backslash empty.$ :* $\backslash empty\; \backslash setminus\; A\; =\; \backslash empty.$ :* $A\; \backslash setminus\; \backslash empty\; =\; A.$ :* $A\; \backslash setminus\; U\; =\; \backslash empty.$ :* If $A\backslash subset\; B$, then $C\backslash setminus\; A\backslash supset\; C\backslash setminus\; B$. :* $A\; \backslash supseteq\; B\; \backslash setminus\; C$ is equivalent to $C\; \backslash supseteq\; B\; \backslash setminus\; A$.Complementary relation

Abinary 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 $X\; \backslash times\; Y.$ The complementary relation $\backslash bar$ is the set complement of $R$ in $X\; \backslash times\; Y.$ The complement of relation $R$ can be written
$$\backslash bar\; \backslash \; =\; \backslash \; (X\; \backslash times\; Y)\; \backslash setminus\; R.$$
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 theLaTeX
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 ...

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

Someprogramming 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) ...

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 ...

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