# User Contributed Dictionary

## English

injective
1. of, relating to, or being an injection

# Extensive Definition

In mathematics, an injective function is a function which associates distinct arguments with distinct values.
An injective function is called an injection, and is also said to be an information-preserving or one-to-one function (the latter is not to be confused with one-to-one correspondence, i.e. a bijective function).
A function f that is not injective is sometimes called many-to-one. (However, this terminology is also sometimes used to mean "single-valued", i.e. each argument is mapped to at most one value.)

## Definition

Let f be a function whose domain is a set A. It is injective if, for all a and b in A such that f(a)=f(b), we have a = b.

## Examples and counter-examples

• For any set X, the identity function on X is injective.
• The function f : R → R defined by f(x) = 2x + 1 is injective.
• The function g : R → R defined by g(x) = x2 is not injective, because (for example) g(1) = 1 = g(−1). However, if g is redefined so that its domain is the non-negative real numbers [0,+∞), then g is injective.
• The exponential function \exp : \mathbb \to \mathbb : x \mapsto \mathrm^x is injective (but not surjective as no value maps to a negative number).
• The natural logarithm function \ln : (0,+\infty) \to \mathbb : x \mapsto \ln is injective.
• The function g : R → R defined by g(x) = x^n - x is not injective, since, for example, g(0) = g(1).
More generally, when X and Y are both the real line R, then an injective function f : R → R is one whose graph is never intersected by any horizontal line more than once.

## Injections can be undone

Functions with left inverses (often called sections) are always injections. That is to say, for f : X → Y, if there exists a function g : Y → X such that, for every x \in X
g(f(x)) = x \, (f can be undone by g)
then f is injective.
Conversely, every injection f with non-empty domain has a left inverse g (in conventional mathematics). Note that g may not be a complete inverse of f because the composition in the other order, f o g, may not be the identity on Y. In other words, a function that can be undone or "reversed", such as f, is not necessarily invertible (bijective). Injections are "reversible" but not always invertible.
Although it is impossible to reverse a non-injective (and therefore information-losing) function, you can at least obtain a "quasi-inverse" of it, that is a multiple-valued function.

## Injections may be made invertible

In fact, to turn an injective function f : X → Y into a bijective (hence invertible) function, it suffices to replace its codomain Y by its actual range J = f(X). That is, let g : X → J such that g(x) = f(x) for all x in X; then g is bijective. Indeed, f can be factored as inclJ,Yog, where inclJ,Yis the inclusion function from J into Y.

## Other properties

• If f and g are both injective, then f o g is injective.
• If g o f is injective, then f is injective (but g need not be).
• f : X → Y is injective if and only if, given any functions g, h : W → X, whenever f o g = f o h, then g = h. In other words, injective functions are precisely the monomorphisms in the category Set of sets.
• If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A. Thus, A can be recovered from its image f(A).
• If f : X → Y is injective and A and B are both subsets of X, then f(A ∩ B) = f(A) ∩ f(B).
• Every function h : W → Y can be decomposed as h = f o g for a suitable injection f and surjection g. This decomposition is unique up to isomorphism, and f may be thought of as the inclusion function of the range h(W) of h as a subset of the codomain Y of h.
• If f : X → Y is an injective function, then Y has at least as many elements as X, in the sense of cardinal numbers.
• If both X and Y are finite with the same number of elements, then f : X → Y is injective if and only if f is surjective.
• Every embedding is injective.

## References

injective in Bulgarian: Инекция
injective in Catalan: Funció injectiva
injective in Czech: Prosté zobrazení
injective in Danish: Injektiv
injective in German: Injektivität
injective in Spanish: Función inyectiva
injective in Esperanto: Enĵeto
injective in French: Injection (mathématiques)
injective in Korean: 단사 함수
injective in Croatian: Injektivna funkcija
injective in Ido: Injektio
injective in Italian: Funzione iniettiva
injective in Hebrew: פונקציה חד חד ערכית
injective in Lithuanian: Injekcija (matematika)
injective in Hungarian: Injektív leképezés
injective in Dutch: Injectie (wiskunde)
injective in Japanese: 単射
injective in Occitan (post 1500): Injeccion (matematicas)
injective in Polish: Funkcja różnowartościowa
injective in Portuguese: Função injectiva
injective in Romanian: Funcţie injectivă
injective in Russian: Инъекция (математика)
injective in Slovak: Prosté zobrazenie
injective in Slovenian: Injektivna preslikava
injective in Serbian: Инјективно пресликавање
injective in Finnish: Injektio
injective in Swedish: Injektiv
injective in Chinese: 单射