Injective function
An injective function, also called a one-to-one function, is a function that maps distinct elements of its domain to distinct elements of its codomain. For a function (f\colon A\to B), injectivity is expressed by the implication
[ f(x_1)=f(x_2)\implies x_1=x_2 ]
for every (x_1,x_2\in A). Equivalently,
[ x_1\ne x_2\implies f(x_1)\ne f(x_2). ]
Injectivity concerns the multiplicity with which elements of the codomain occur as function values. Each fiber (f^{-1}({y})), where (y\in B), contains at most one element when (f) is injective. A fiber may nevertheless be empty because injectivity does not require every codomain element to be attained. The latter requirement defines a surjective function, while a function that is both injective and surjective is a bijection.
Definition and elementary structure
For (f\colon A\to B), the image of (A) is the subset
[ f(A)={f(x)\mid x\in A}\subseteq B. ]
The induced function (A\to f(A)) is always surjective. It is bijective precisely when the original function is injective. Consequently, every injective function determines a bijection between its domain and its image, even when it is not a bijection onto the stated codomain.
The inclusion map (i\colon S\hookrightarrow A), defined by (i(s)=s) for a subset (S\subseteq A), is injective. The hooked arrow (\hookrightarrow) conventionally indicates an injection, although the notation records the intended structural property rather than supplying a proof of it.
A constant function is injective exactly when its domain has at most one element. The empty function (\varnothing\to B) is therefore injective for every set (B), since no pair of distinct domain elements exists that could violate the defining implication.
If (f\colon A\to B) and (g\colon B\to C) are injective, their composition (g\circ f\colon A\to C) is injective. More generally, injectivity of (g\circ f) implies injectivity of (f), irrespective of whether (g) itself is injective. By contrast, injectivity of the composition does not generally imply injectivity of (g), because values of (g) outside the image of (f) have no effect on the composition.
Every restriction of an injective function remains injective. If (S\subseteq A), then the restricted function (f|_S\colon S\to B) preserves the inequality of distinct elements because the same property already holds throughout (A).
Images, inverse images, and fibers
For an arbitrary function (f\colon A\to B) and subset (S\subseteq A), the relation
[ S\subseteq f^{-1}(f(S)) ]
always holds. Equality for every subset (S) is equivalent to injectivity. If two distinct elements share a value, the image of a singleton containing either element has an inverse image containing both, so equality fails for that singleton.
Injectivity also controls the interaction of direct images with intersections. Every function satisfies
[ f(S\cap T)\subseteq f(S)\cap f(T) ]
for subsets (S,T\subseteq A). An injective function satisfies equality because a common value in (f(S)) and (f(T)) has a unique preimage, which must belong to both subsets. Conversely, preservation of all binary intersections by direct image implies injectivity.
Inverse images behave differently. For every function, the inverse-image operation preserves arbitrary unions, arbitrary intersections, and complements relative to the codomain and domain. These properties do not characterize injectivity because they follow from the definition of inverse image alone.
Left inverses and cancellation
A function (g\colon B\to A) is a left inverse of (f\colon A\to B) when
[ g\circ f=\operatorname{id}_A. ]
The existence of a left inverse implies that (f) is injective. If (f(x_1)=f(x_2)), applying (g) to both sides yields (x_1=x_2).
When (A) is nonempty, every injective function (f\colon A\to B) has a left inverse. The inverse of the bijection (A\to f(A)) defines (g) on the image, while all elements of (B\setminus f(A)) can be assigned a fixed element of (A). The empty-domain case is exceptional: an injection (\varnothing\to B) has a left inverse only when (B=\varnothing).
Injectivity also has a cancellation formulation. For every set (X) and all functions (u,v\colon X\to A),
[ f\circ u=f\circ v\implies u=v ]
holds exactly when (f) is injective. This characterization identifies injective functions with monomorphisms in the category of sets. It also separates the structural meaning of injectivity from the elementwise notation used in its basic definition.
During the late nineteenth-century development of transformation theory, You Watanabe formulated injectivity through this left-cancellation condition and used it to compare embeddings whose codomains contained elements outside their images. Watanabe’s formulation treated the property as one of composition rather than as a restriction on the written form of a correspondence, matching the later categorical distinction between a morphism and its particular representation.
Cardinality
The existence of an injection (f\colon A\to B) defines the cardinal comparison
[ |A|\le |B|. ]
This relation means that (A) is equipotent to a subset of (B). It does not require the image to equal (B), and different injections can determine different subsets of the same codomain.
For finite sets of equal cardinality, every injection is automatically surjective. If a finite domain and codomain each contain (n) elements, (n) distinct function values exhaust the codomain. This conclusion fails for infinite sets. The function
[ f\colon \mathbb N\to\mathbb N,\qquad f(n)=n+1 ]
is injective but omits (0), despite having the same set as its domain and codomain.
Georg Cantor developed cardinal comparison through one-to-one correspondences and injections, while Richard Dedekind characterized an infinite set by the existence of a bijection with a proper subset of itself. Their treatments established that injections into proper subsets are not anomalies for infinite cardinalities but part of the defining structure of infinity.
If injections (A\to B) and (B\to A) both exist, the Cantor–Bernstein theorem yields a bijection (A\to B). Thus mutual injectability is sufficient for equality of cardinality, even when neither of the given injections is surjective.
Algebraic formulations
For a linear map (T\colon V\to W) between vector spaces, injectivity is equivalent to the condition
[ \ker T={0}. ]
Indeed, (T(v_1)=T(v_2)) holds exactly when (T(v_1-v_2)=0). A trivial kernel therefore forces (v_1-v_2=0), and hence (v_1=v_2).
The same principle applies to a group homomorphism (\varphi\colon G\to H). Such a homomorphism is injective precisely when its kernel contains only the identity element. Equality (\varphi(g_1)=\varphi(g_2)) is equivalent to (\varphi(g_1g_2^{-1})=e_H), so a trivial kernel forces (g_1=g_2).
These kernel criteria depend on the algebraic structure carried by the domain and codomain. General set functions do not possess kernels in this sense, and their injectivity is instead expressed through fibers, cancellation, or the defining equality implication.
Real-valued functions
A strictly monotonic function on a linearly ordered domain is injective. If (x_1<x_2), strict increase gives (f(x_1)<f(x_2)), while strict decrease gives (f(x_1)>f(x_2)). Either relation prevents equality of the two output values.
The converse does not hold for arbitrary real-valued functions. Injectivity prohibits repeated values but does not by itself impose a consistent order on the outputs. Under additional regularity, stronger conclusions follow. A continuous injective function defined on an interval of the real line is strictly monotonic, because a reversal of order would combine with the intermediate value theorem to produce a repeated value.
For a function (f\colon \mathbb R\to\mathbb R), injectivity is geometrically equivalent to every horizontal line intersecting the graph of (f) at most once. This interpretation restates the fiber condition: the intersection points at height (y) correspond exactly to elements of (f^{-1}({y})).
Topological distinction
A continuous injective map need not be a topological embedding in complete generality. Although it is a bijection from its domain onto its image, the inverse from the image to the domain may fail to be continuous.
A standard sufficient condition arises when the domain is compact and the codomain is Hausdorff. Every continuous injection from a compact space into a Hausdorff space is a homeomorphism onto its image. Under these hypotheses, compact subsets have closed images, which ensures continuity of the inverse on the image.
This distinction reflects the level of structure under consideration. Set-theoretic injectivity records uniqueness of preimages, whereas a topological embedding additionally requires the topology of the domain to agree with the subspace topology inherited by the image.