# User Contributed Dictionary

### Adjective

injective- of, relating to, or being an injection

#### Derived terms

#### Translations

- Swedish: injektiv

# 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)

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.

## See also

## Notes

## References

- The Elements of Real Analysis , p. 17 ff.
- Naive Set Theory , p. 38 ff.

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