Bijection, injection and surjection

Injective composition: the second function need not be injective.

A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images. An injective function is an injection.[1] The formal definition is the following.

The function ${\displaystyle f\colon X\to Y}$ is injective, if for all ${\displaystyle x,x'\in X}$, ${\displaystyle f(x)=f(x')\Rightarrow x=x'.}$[2][3][4]

The following are some facts related to injections:

• A function ${\displaystyle f\colon X\to Y}$ is injective if and only if ${\displaystyle X}$ is empty or ${\displaystyle f}$ is left-invertible; that is, there is a function ${\displaystyle g\colon f(X)\to X}$ such that ${\displaystyle g\circ f=}$ identity function on X. Here, ${\displaystyle f(X)}$ is the image of ${\displaystyle f}$.
• Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image. More precisely, every injection ${\displaystyle f\colon X\to Y}$ can be factored as a bijection followed by an inclusion as follows. Let ${\displaystyle f_{R}\colon X\rightarrow f(X)}$ be ${\displaystyle f}$ with codomain restricted to its image, and let ${\displaystyle i\colon f(X)\to Y}$ be the inclusion map from ${\displaystyle f(X)}$ into ${\displaystyle Y}$. Then ${\displaystyle f=i\circ f_{R}}$. A dual factorization is given for surjections below.
• The composition of two injections is again an injection, but if ${\displaystyle g\circ f}$ is injective, then it can only be concluded that ${\displaystyle f}$ is injective (see figure).
• Every embedding is injective.

Surjective composition: the first function need not be surjective.

A function is surjective or onto if each element of the codomain is mapped to by at least one element of the domain. In other words, each element of the codomain has a non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection.[1] The formal definition is the following.

The function ${\displaystyle f\colon X\to Y}$ is surjective, if for all ${\displaystyle y\in Y}$, there is ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=y.}$[2][3][4]

The following are some facts related to surjections:

• A function ${\displaystyle f\colon X\to Y}$ is surjective if and only if it is right-invertible, that is, if and only if there is a function ${\displaystyle g\colon Y\to X}$ such that ${\displaystyle f\circ g=}$ identity function on ${\displaystyle Y}$. (This statement is equivalent to the axiom of choice.)
• By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection from a quotient set of its domain to its codomain. More precisely, the preimages under f of the elements of the image of ${\displaystyle f}$ are the equivalence classes of an equivalence relation on the domain of ${\displaystyle f}$, such that x and y are equivalent if and only they have the same image under ${\displaystyle f}$. As all elements of any one of these equivalence classes are mapped by ${\displaystyle f}$on the same element of the codomain, this induces a bijection between the quotient set by this equivalence relation (the set of the equivalence classes) and the image of ${\displaystyle f}$ (which is its codomain when ${\displaystyle f}$ is surjective). Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of ${\displaystyle f}$.
• The composition of two surjections is again a surjection, but if ${\displaystyle g\circ f}$ is surjective, then it can only be concluded that ${\displaystyle g}$ is surjective (see figure).

Bijective composition: the first function need not be surjective and the second function need not be injective.

A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence. A function is bijective if and only if every possible image is mapped to by exactly one argument.[1] This equivalent condition is formally expressed as follows:

The function ${\displaystyle f\colon X\to Y}$ is bijective, if for all ${\displaystyle y\in Y}$, there is a unique ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=y.}$[2][3][4]

The following are some facts related to bijections:

• A function ${\displaystyle f\colon X\to Y}$ is bijective if and only if it is invertible, that is, there is a function ${\displaystyle g\colon Y\to X}$ such that ${\displaystyle g\circ f=}$ identity function on X and ${\displaystyle f\circ g=}$ identity function on ${\displaystyle Y}$. This function maps each image to its unique preimage.
• The composition of two bijections is again a bijection, but if ${\displaystyle g\circ f}$ is a bijection, then it can only be concluded that ${\displaystyle f}$ is injective and ${\displaystyle g}$ is surjective (see the figure at right and the remarks above regarding injections and surjections).
• The bijections from a set to itself form a group under composition, called the symmetric group.

It is important to specify the domain and codomain of each function, since by changing these, functions which appear to be the same may have different properties.

Injective and surjective (bijective)

The identity function id_X for every non-empty set X, and thus specifically ${\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto x.}$

${\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto x^{2}}$, and thus also its inverse ${\displaystyle \mathbf {R} ^{+}\to \mathbf {R} ^{+}:x\mapsto {\sqrt {x}}.}$

The exponential function ${\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} ^{+}:x\mapsto \mathrm {e} ^{x}}$ (that is, the exponential function with its codomain restricted to its image), and thus also its inverse the natural logarithm ${\displaystyle \ln \colon \mathbf {R} ^{+}\to \mathbf {R} :x\mapsto \ln {x}.}$

Injective and non-surjective

The exponential function ${\displaystyle \exp \colon \mathbf {R} \to \mathbf {R} :x\mapsto \mathrm {e} ^{x}.}$

Non-injective and surjective

${\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto (x-1)x(x+1)=x^{3}-x.}$

${\displaystyle \mathbf {R} \to [-1,1]:x\mapsto \sin(x).}$

Non-injective and non-surjective

${\displaystyle \mathbf {R} \to \mathbf {R} :x\mapsto \sin(x).}$

• For every function f, subset X of the domain and subset Y of the codomain, X ⊂ f⁻¹(f(X)) and f(f⁻¹(Y)) ⊂ Y. If f is injective, then X = f⁻¹(f(X)), and if f is surjective, then f(f⁻¹(Y)) = Y.
• For every function h : X → Y, one can define a surjection H : X → h(X) : x → h(x) and an injection I : h(X) → Y : y → y. It follows that ${\displaystyle h=I\circ H}$. This decomposition is unique up to isomorphism.

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.[5]

The Oxford English Dictionary records the use of the word injection as a noun by S. Mac Lane in Bulletin of the American Mathematical Society (1950), and injective as an adjective by Eilenberg and Steenrod in Foundations of Algebraic Topology (1952).[6]

However, it was not until the French Bourbaki group coined the injective-surjective-bijective terminology (both as nouns and adjectives) that they achieved widespread adoption.[7]

• Horizontal line test
• Injective module
• Permutation

• Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms.