# Surjective function

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In mathematics, a surjective function (also known as surjection, or onto function) is a function f such that every element y can be mapped from element x so that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. [1] [2] It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

## Contents

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, [3] [4] a group of mainly French 20th-century mathematicians 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 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 assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

## Definition

A surjective function is a function whose image is equal to its codomain. Equivalently, a function ${\displaystyle f}$ with domain ${\displaystyle X}$ and codomain ${\displaystyle Y}$ is surjective if for every ${\displaystyle y}$ in ${\displaystyle Y}$ there exists at least one ${\displaystyle x}$ in ${\displaystyle X}$ with ${\displaystyle f(x)=y}$. [1] Surjections are sometimes denoted by a two-headed rightwards arrow ( U+21A0RIGHTWARDS TWO HEADED ARROW), [5] as in ${\displaystyle f\colon X\twoheadrightarrow Y}$.

Symbolically,

If ${\displaystyle f\colon X\rightarrow Y}$, then ${\displaystyle f}$ is said to be surjective if and only if
${\displaystyle \forall y\in Y,\,\exists x\in X,\;\;f(x)=y}$. [2] [6]

## Examples

• For any set X, the identity function idX on X is surjective.
• The function f : Z → {0, 1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1) is surjective.
• The function f : RR defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number y, we have an x such that f(x) = y: such an appropriate x is (y − 1)/2.
• The function f : RR defined by f(x) = x3 − 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x3 − 3xy = 0, and every cubic polynomial with real coefficients has at least one real root. However, this function is not injective (and hence not bijective), since, for example, the pre-image of y = 2 is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)
• The function g : RR defined by g(x) = x2 is not surjective, since there is no real number x such that x2 = −1. However, the function g : RR≥0 defined by g(x) = x2 (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 x2 = y.
• The natural logarithm function ln : (0, +∞) → R 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, 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 is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the general linear group of degree n (that is, the group of all n×n invertible matrices). Under this definition, the matrix exponential is surjective for complex matrices, although still not surjective for real matrices.
• The projection from a cartesian product A × B 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 is bijective if and only if it is both surjective and injective.

If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, but rather a property of the mapping. [7] 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 g : YX is said to be a right inverse of the function f : XY 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 the composition fog of g and f in that order is the identity function on the domain Y of g. The function g need not be a complete inverse of f because the composition in the other order, gof, 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.

If f : XY is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimage f −1(B).

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.

### Surjections as epimorphisms

A function f : XY is surjective if and only if it is right-cancellative: [8] given any functions g,h : YZ, whenever gof = hof, then g = h. This property is formulated in terms of functions and their composition and can be generalized to the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms are called epimorphisms. Specifically, surjective functions are precisely the epimorphisms in the category of sets. 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 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 its function graph. 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

The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f : XY is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (The proof appeals to the axiom of choice to show that a function g : YX satisfying f(g(y)) = y for all y in Y exists. g is easily seen to be injective, thus the formal definition of |Y| ≤ |X| is satisfied.)

Specifically, if both X and Y are finite with the same number of elements, then f : XY is surjective if and only if f is injective.

Given two sets X and Y, the notation X*Y 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 X*Y and Y*X together imply that |Y| = |X|, a variant of the Schröder–Bernstein theorem.

### Composition and decomposition

The composition 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 fog is surjective. Conversely, if fog is surjective, then f is surjective (but g, the function applied first, need not be). These properties generalize from surjections in the category of sets to any epimorphisms in any category.

Any function can be decomposed into a surjection and an injection: For any function h : XZ there exist a surjection f : XY and an injection g : YZ such that h = gof. To see this, define Y to be the set of preimages h−1(z) where z is in h(X). These preimages are disjoint and partition 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 a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, every surjection f : AB can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : AA/~ be the projection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).

## Space of surjections

Given fixed A and B, one can form the set of surjections AB. The cardinality of this set is one of the twelve aspects of Rota's Twelvefold way, and is given by ${\textstyle |B|!{\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}$, where ${\textstyle {\begin{Bmatrix}|A|\\|B|\end{Bmatrix}}}$ denotes a Stirling number of the second kind.

## Related Research Articles

In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: XY is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one-to-one correspondence must not be confused with one-to-one function.

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.

In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: VV, and an endomorphism of a group G is a group homomorphism f: GG. In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : GH such that for all u and v in G it holds that

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, the inverse function of a function f is a function that undoes the operation of f. The inverse of f exists if and only if f is bijective, and if it exists, is denoted by

In mathematics, a partial functionf from a set X to a set Y is a function from a subset S of X to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition of f. If S equals X, that is, if f is defined on every element in X, then f is said to be total.

In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : RS such that f is:

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. (Equivalently, x1x2 implies f(x1) ≠ f(x2) in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

In mathematics, the concept of an inverse element generalises the concepts of opposite and reciprocal of numbers.

In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set Y in the notation f: XY. The term range is sometimes ambiguously used to refer to either the codomain or image of a function.

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local structure. If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In mathematics, the image of a function is the set of all output values it may produce.

In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both properties is called a full and faithful functor.

In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other.

In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

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