In

_{''X''} on ''X'' is surjective.
* The function defined by ''f''(''n'') = ''n'' mod 2 (that is, even ^{3} − 3''x'' is surjective, because the pre-image of any ^{3} − 3''x'' − ''y'' = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not ^{2} is ''not'' surjective, since there is no real number ''x'' such that ''x''^{2} = −1. However, the function defined by (with the restricted codomain) ''is'' surjective, since for every ''y'' in the nonnegative real codomain ''Y'', there is at least one ''x'' in the real domain ''X'' such that ''x''^{2} = ''y''.
* The

bijective
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 ...

if and only if it is both surjective and

^{ −1}(''B'')) = ''B''. Thus, ''B'' can be recovered from its
File:Surjective composition.svg, Surjective composition: the first function need not be surjective.
File:Bijection.svg, Another surjective function. (This one happens to be a

composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

of surjective functions is always surjective: If ''f'' and ''g'' are both surjective, and the codomain of ''g'' is equal to the domain of ''f'', then is surjective. Conversely, if is surjective, then ''f'' is surjective (but ''g'', the function applied first, need not be). These properties generalize from surjections in the

_{''P''} : ''A''/~ → ''B'' be the well-defined function given by ''f''_{''P''}( 'x''sub>~) = ''f''(''x''). Then ''f'' = ''f''_{''P''} o ''P''(~).

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 their changes (cal ...

, a surjective function (also known as surjection, or onto function) is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

that maps an element to every element ; that is, for every , there is an such that . In other words, every element of the function's codomain
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 ...

is the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment
Environment most often refers to:
__NOTOC__
* Natural environment, all living and non- ...

of one element of its domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

. It is not required that be unique; the function may map one or more elements of to the same element of .
The term ''surjective'' and the related terms ''injective
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 ...

'' and ''bijective
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 ...

'' were introduced by Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended ...

, a group of mainly 20th-century mathematician
A mathematician is someone who uses an extensive knowledge of 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 ( ...

s who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment
Environment most often refers to:
__NOTOC__
* Natural environment, all living and non- ...

of the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse, and every function with a right inverse is necessarily a surjection. The composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.
Definition

A surjective function is afunction
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

whose image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment
Environment most often refers to:
__NOTOC__
* Natural environment, all living and non- ...

is equal to its codomain
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 ...

. Equivalently, a function $f$ with domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

$X$ and codomain $Y$ is surjective if for every $y$ in $Y$ there exists at least one $x$ in $X$ with $f(x)=y$. Surjections are sometimes denoted by a two-headed rightwards arrow (), as in $f\backslash colon\; X\backslash twoheadrightarrow\; Y$.
Symbolically,
:If $f\backslash colon\; X\; \backslash rightarrow\; Y$, then $f$ is said to be surjective if
:$\backslash forall\; y\; \backslash in\; Y,\; \backslash ,\; \backslash exists\; x\; \backslash in\; X,\; \backslash ;\backslash ;\; f(x)=y$.
Examples

* For any set ''X'', theidentity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

idinteger
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...

s are mapped to 0 and odd integers to 1) is surjective.
* The function defined by ''f''(''x'') = 2''x'' + 1 is surjective (and even bijective
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 ...

), because for every 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 ...

''y'', we have an ''x'' such that ''f''(''x'') = ''y'': such an appropriate ''x'' is (''y'' − 1)/2.
* The function defined by ''f''(''x'') = ''x''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 ...

''y'' is the solution set of the cubic polynomial equation ''x''injective
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 ...

(and hence not bijective
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 ...

), since, for example, the pre-image of ''y'' = 2 is . (In fact, the pre-image of this function for every ''y'', −2 ≤ ''y'' ≤ 2 has more than one element.)
* The function defined by ''g''(''x'') = ''x''natural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...

function is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the exponential function
The exponential function is a mathematical function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of ...

, if defined with the set of real numbers as the domain, is not surjective (as its range is the set of positive real numbers).
* The matrix exponential
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 gene ...

is not surjective when seen as a map from the space of all ''n''×''n'' matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

to itself. It is, however, usually defined as a map from the space of all ''n''×''n'' matrices to the general linear group
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 ...

of degree ''n'' (that is, the group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

of all ''n''×''n'' invertible matrices
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and t ...

). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
* The projection from a cartesian product
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 ...

to one of its factors is surjective, unless the other factor is empty.
* In a 3D video game, vectors are projected onto a 2D flat screen by means of a surjective function.
Properties

A function isinjective
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 ...

.
If (as is often done) a function is identified with its graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

, then surjectivity is not a property of the function itself, but rather a property of the mapping
Mapping may refer to:
* Mapping (cartography), the process of making a map
* Mapping (mathematics), a synonym for a mathematical function and its generalizations
** Mapping (logic), a synonym for functional predicate
Types of mapping
* Animated ...

. This is, the function together with its codomain. Unlike injectivity, surjectivity cannot be read off of the graph of the function alone.
Surjections as right invertible functions

The function is said to be a right inverse of the function if ''f''(''g''(''y'')) = ''y'' for every ''y'' in ''Y'' (''g'' can be undone by ''f''). In other words, ''g'' is a right inverse of ''f'' if thecomposition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

of ''g'' and ''f'' in that order is the identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

on the domain ''Y'' of ''g''. The function ''g'' need not be a complete inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when add ...

of ''f'' because the composition in the other order, , may not be the identity function on the domain ''X'' of ''f''. In other words, ''f'' can undo or "''reverse''" ''g'', but cannot necessarily be reversed by it.
Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has a right inverse is equivalent to the axiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

.
If is surjective and ''B'' is a 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 ''Y'', then ''f''(''f''preimage
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). ...

.
For example, in the first illustration above, there is some function ''g'' such that ''g''(''C'') = 4. There is also some function ''f'' such that ''f''(4) = ''C''. It doesn't matter that ''g''(''C'') can also equal 3; it only matters that ''f'' "reverses" ''g''.
bijection
In , a bijection, bijective function, one-to-one correspondence, or invertible function, is a between the elements of two , where each element of one set is paired with exactly one element of the other set, and each element of the other set is p ...

)
File:Injection.svg, A non-surjective function. (This one happens to be an injection
Injection or injected may refer to:
Science and technology
* Injection (medicine)
An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

)
Surjections as epimorphisms

A function is surjective if and only if it isright-cancellative
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 ...

: given any functions , whenever ''g'' o ''f'' = ''h'' o ''f'', then ''g'' = ''h''. This property is formulated in terms of functions and their composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

and can be generalized to the more general notion of the morphism
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 gener ...

s of a category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

and their composition. Right-cancellative morphisms are called epimorphism
220px
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...

s. Specifically, surjective functions are precisely the epimorphisms in the category of setsIn the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

. The prefix ''epi'' is derived from the Greek preposition ''ἐπί'' meaning ''over'', ''above'', ''on''.
Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse ''g'' of a morphism ''f'' is called a section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...

of ''f''. A morphism with a right inverse is called a split epimorphism.
Surjections as binary relations

Any function with domain ''X'' and codomain ''Y'' can be seen as a left-total and right-unique binary relation between ''X'' and ''Y'' by identifying it with itsfunction graph
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). ...

. A surjective function with domain ''X'' and codomain ''Y'' is then a binary relation between ''X'' and ''Y'' that is right-unique and both left-total and right-total.
Cardinality of the domain of a surjection

Thecardinality
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 ...

of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If is a surjective function, then ''X'' has at least as many elements as ''Y'', in the sense of cardinal number
150px, Aleph null, the smallest infinite cardinal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...

s. (The proof appeals to the axiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

to show that a function
satisfying ''f''(''g''(''y'')) = ''y'' for all ''y'' in ''Y'' exists. ''g'' is easily seen to be injective, thus the formal definition
Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements (form
Form is the shape, visual appearance, or :wikt:configuration, configuration of an object. In a wider sense, the form is the ...

of , ''Y'', ≤ , ''X'', is satisfied.)
Specifically, if both ''X'' and ''Y'' are finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

with the same number of elements, then is surjective if and only if ''f'' is injective
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 ...

.
Given two sets ''X'' and ''Y'', the notation is used to say that either ''X'' is empty or that there is a surjection from ''Y'' onto ''X''. Using the axiom of choice one can show that and together imply that , ''Y'', = , ''X'', , a variant of the Schröder–Bernstein theoremIn set theory, the Schröder–Bernstein theorem states that, if there exist injective functions and between the Set (mathematics), sets and , then there exists a bijection, bijective function .
In terms of the cardinality of the two sets, this c ...

.
Composition and decomposition

Thecategory of setsIn the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

to any epimorphism
220px
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...

s in any category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

.
Any function can be decomposed into a surjection and an injection
Injection or injected may refer to:
Science and technology
* Injection (medicine)
An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

: For any function there exist a surjection and an injection such that ''h'' = ''g'' o ''f''. To see this, define ''Y'' to be the set of preimage
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). ...

s where ''z'' is in . These preimages are and ''X''. Then ''f'' carries each ''x'' to the element of ''Y'' which contains it, and ''g'' carries each element of ''Y'' to the point in ''Z'' to which ''h'' sends its points. Then ''f'' is surjective since it is a projection map, and ''g'' is injective by definition.
Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on aquotient
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...

of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection can be factored as a projection followed by a bijection as follows. Let ''A''/~ be the equivalence class
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 an ...

es of ''A'' under the following equivalence relation
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 ...

: ''x'' ~ ''y'' if and only if ''f''(''x'') = ''f''(''y''). Equivalently, ''A''/~ is the set of all preimages under ''f''. Let ''P''(~) : ''A'' → ''A''/~ be the projection map
In mathematics, a projection is a mapping of a Set (mathematics), set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for function composition, mapping composition (or, in other words, which is idem ...

which sends each ''x'' in ''A'' to its equivalence class 'x''sub>~, and let ''f''Space of surjections

Given fixed and , one can form the set of surjections . Thecardinality
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 ...

of this set is one of the twelve aspects of Rota's Twelvefold way
In combinatorics
Combinatorics is an area of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, ...

, and is given by $,\; B,\; !\backslash begin,\; A,\; \backslash \backslash ,\; B,\; \backslash end$, where $\backslash begin,\; A,\; \backslash \backslash ,\; B,\; \backslash end$ denotes a Stirling number of the second kind
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 ...

.
See also

*Bijection, injection and surjection
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). ...

* Cover (algebra)
*Covering map
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). ...

*Enumeration
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and r ...

*Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in English in the Commonwealth of Nations, Commonwealth English: fibre bundle) is a Space (mathematics), space that is ''locally'' a product space, but ''globally'' may have a dif ...

*Index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...

*Section (category theory)
In category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

References

Further reading

* {{Cite book , title=Theory of Sets , last=Bourbaki , first=N. , author-link=Nicolas Bourbaki , year=2004 , orig-year=1968 , publisher=Springer , isbn=978-3-540-22525-6 , doi=10.1007/978-3-642-59309-3 , lccn=2004110815 , series= , volume=1 , url=https://books.google.com/books?id=7eclBQAAQBAJ&pg=PR1 , ref=bourbaki Functions and mappingsBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...

Mathematical relations
Types of functions