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][2] The formal definition is the following.

The function ${\displaystyle f\colon X\to Y}$ is injective, if for all ${\displaystyle x,x'\in X}$, ${\displaystyle f(x)=f(x')\Rightarrow x=x'.}$[3][4][5]

The following are some facts related to injections:

• A function f : X → Y is injective if and only if X is empty or f is left-invertible; that is, there is a function g : f(X) → X such that g o f = identity function on X. Here, f(X) is the image of f.
• Since every function is surjective when its codomain is restricted to its image, every injection induces a bijection onto its image.[1] More precisely, every injection f : X → Y can be factored as a bijection followed by an inclusion as follows. Let f_R : X → f(X) be f with codomain restricted to its image, and let i : f(X) → Y be the inclusion map from f(X) into Y. Then f = i o f_R. A dual factorisation is given for surjections below.
• The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective (see figure).
• Every embedding is injective.

Surjective composition: the first function need not be surjective.

A function is surjective (onto) if each possible image is mapped to by at least one argument. In other words, each element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection.[1][2] 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 ${\displaystyle f(x)=y.}$[3][4][5]

The following are some facts related to surjections:

• A function f : X → Y is surjective if and only if it is right-invertible, that is, if and only if there is a function g: Y → X such that f o g = identity function on 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 defined on a quotient of its domain. More precisely, every surjection f : X → Y can be factored as a non-bijection followed by a bijection as follows. Let X/~ be the equivalence classes of X under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, X/~ is the set of all preimages under f. Let P(~) : X → X/~ be the projection map which sends each x in X to its equivalence class [x]_~, and let f_P : X/~ → Y be the well-defined function given by f_P([x]_~) = f(x). Then f = f_P o P(~). A dual factorisation is given for injections above.
• The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concluded that 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 a bijection (one-to-one correspondence).[1] A function is bijective if and only if every possible image is mapped to by exactly one argument.[2] This equivalent condition is formally expressed as follow.

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 ${\displaystyle f(x)=y.}$[3][4][5]

The following are some facts related to bijections:

• A function f : X → Y is bijective if and only if it is invertible, that is, there is a function g: Y → X such that g o f = identity function on X and f o g = identity function on Y. This function maps each image to its unique preimage.
• The composition of two bijections is again a bijection, but if g o f is a bijection, then it can only be concluded that f is injective and 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.[6]

The injective-surjective-bijective terminology (both as nouns and adjectives) was originally coined by the French Bourbaki group, before their 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.