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...

g\circ f} is injective, then it can only be concluded that f {\displaystyle f} is injective (see figure). Every embedding is injective. A function is surjective...

continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function f : X → Y {\displaystyle f:X\to...

partial function which is injective. An injective partial function may be inverted to an injective partial function, and a partial function which is...

function f itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion...

natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in...

be unique; the function f may map one or more elements of X to the same element of Y. The term surjective and the related terms injective and bijective...

g(y)=g(f(h(y))=h(y)} . A function has a two-sided inverse if and only if it is bijective. A bijective function f is injective, so it has a left inverse...

has the division by two as its inverse function. A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element...

up inject, injected, injecting, injection, or injections in Wiktionary, the free dictionary. Injection or injected may refer to: Injective function, a...

equivalent to counting injective functions N → X. Counting n-combinations of X is equivalent to counting injective functions N → X up to permutations...

element. An empty function is always injective. If X is not the empty set, then f is injective if and only if there exists a function g : Y → X {\displaystyle...

Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but...

element x in the domain X. The identity function on X is clearly an injective function as well as a surjective function (its codomain is also its range), so...

A to B that is not injective, then no surjection from A to B is injective. In fact no function of any kind from A to B is injective. This is not true for...

analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. The function f : z ↦ 2 z + z 2 {\displaystyle...

space, is a local homeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle f} is injective on some neighborhood of A {\displaystyle...

(respectively, a continuous map, etc.). The restriction of the non-injective function f : R → R ,   x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb {R}...

composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows...

if and only if any of the following conditions hold: There is no injective function (hence no bijection) from X to the set of natural numbers. X is nonempty...

an injective function. Perfect hash functions may be used to implement a lookup table with constant worst-case access time. A perfect hash function can...

that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of...

cardinality of X (with the bijection definition of cardinality and the injective function order). (If we restrict to cardinal numbers of well-orderable sets...

the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Formally, a set...

smooth immersion is a locally injective function, while invariance of domain guarantees that any continuous injective function between manifolds of equal...

whether a function is injective (i.e., one-to-one). A horizontal line is a straight, flat line that goes from left to right. Given a function f : R → R...

objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds...

can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are...

Simple path may refer to: Simple curve, a continuous injective function from an interval in the set of real numbers R {\displaystyle \mathbb {R} } to R...

functions. A bi-Lipschitz function is a Lipschitz function φ : U → Rn which is injective and whose inverse function φ−1 : φ(U) → U is also Lipschitz. By Rademacher's...