Вы находитесь на странице: 1из 8

# Injective function

"One-to-one" redirects here. For other uses, see One-to-one (disambiguation).
"Injective" redirects here. For injective modules, see Injective module.

## A non-injective function (this one happens to be a surjection)

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct
elements of its domain to the same element of its codomain. In other words, every element of the
function's codomain is mapped to by at most one element of its domain. If in addition all of the
elements in the codomain are in fact mapped to by some element of the domain, then the function is
said to be bijective (see figures).

An injective function is called an injection, and is also said to be a one-to-one function (not to be
confused with one-to-one correspondence, i.e. a bijective function). Occasionally, an injective function
from X to Y is denoted f: X ֌ Y, using an arrow with a barbed tail. Alternately, it may be
denoted YXusing a notation derived from that used for falling factorial powers, since if X and Y are
finite sets with respectively x and y elements, the number of injections X ֌ Y is yx (see the twelvefold
way).

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; this is
the case for any function, but is used to stress the opposition with multi-valued functions, which are not
true functions.)

Contents
[hide]

1 Definition

2 Examples

## 4 Injections may be made invertible

5 Other properties

7 Notes

8 References

Definition

Let f be a function whose domain is a set A. The function f is injective if for all a and b in A, if f(a)
=f(b), then a = b; that is, f(a) = f(b) implies a = b. Equivalently, if a ≠ b, then f(a) ≠ f(b).

Examples
 For any set X and any subset S of X the inclusion map S → X (which sends any
element s of S to itself) is injective. In particular the identity function X → X is always injective (and
in fact bijective).

##  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 : R → R defined by exp(x) = ex is injective (but not surjective as
no value maps to a negative number).

##  The natural logarithm function ln : (0, ∞) → R defined by x ֏ ln x is injective.

 The function g : R → R defined by g(x) = xn − 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. This principle is referred to as
the horizontal line test.

## Injections can be undone

Functions with left inverses are always injections. That is, given f : X → Y, if there is a
function g : Y → X such that, for every x X

## then f is injective. In this case, f is called a section of g and g is called a retraction of f.

Conversely, every injection f with non-empty domain has a left inverse g (in conventional
mathematics[1]). Note that g may not be a completeinverse of f because the composition in the
other order, f ◦ 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,

one 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,Y ◦ g, where inclJ,Y is
the inclusion function from J into Y.
Other properties

##  If g ◦ 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, wheneverf ◦ g = f ◦ h,
then g = h. In other words, injective functions are precisely themonomorphisms 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 ◦ g for a suitable injectionf 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. In particular, if, in addition, there is an injection from Y to X,
then X and Y has the same cardinal number. (This is known as the Cantor–Bernstein–
Schroeder theorem.)

 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 (in which case they arebijective).

##  An injective function which is a homomorphism between two algebraic structures is

an embedding.

Surjective function
(Redirected from Surjection)
"Onto" redirects here. For other uses, see wikt:onto.

In mathematics, a function is said to be surjective or onto if its image is equal to its codomain. A
function f: X → Y is surjective if and only iffor every y in the codomain Y there is at least one x in
the domain X such that f(x) = y. A surjective function is called a surjection. Surjections are sometimes
denoted by a two-headed rightwards arrow, as in f: X ։ Y.

The term surjective and the related terms injective and bijective were introduced by Nicolas
Bourbaki,[1] a group of mainly French 20th-century mathematicians who wrote a series of books
presenting an exposition of modern advanced mathematics, beginning in 1935. The
French prefix sur means over or above and relates to the fact that the image of the domain of a
surjective function completely covers the function's codomain.

Contents
[hide]

1 Examples

2 Properties

4 Notes

5 References

Examples

## For any set X, the identity function idX on X is surjective.

The function f: Z → {0,1} defined by f(n) = n mod 2 and mapping even integers to 0 and odd integers
to 1 is surjective.

The function f: R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real
number y we have an x such that f(x) =y: an appropriate x is (y − 1)/2.

The function g: R → R defined by g(x) = x2 is not surjective, because there is no real number x such
that x2 = −1. However, the functiong: R → R+ defined by g(x) = x2 (with restricted
codomain) is surjective because for every y in the positive real codomain Y there is at least one x in
the real domain X such that x2 = y.

The natural logarithm function ln: (0,+∞) → R is a surjective and even bijective mapping from the set of
positive real numbers to the set of all real numbers. Its inverse, the exponential function, is not
surjective as its range is the set of positive real numbers and its domain is usually defined to be the set
of all real numbers. The matrix exponential is not surjective when seen as a map from the space of
all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to
the general linear group of degree n, i.e. the group of alln×n invertible matrices. Under this definition
the matrix exponential is surjective.

Properties

## Surjections as right invertible functions

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for
every y in Y (g can be undone by f). In other words, g is a right inverse of f if
the composition f o g of g and f in that order is the identity function on the domain Y of g. The
function g need not be a complete inverse of f because the composition in the other order, g o f, may
not be the identity function on the domain X of f. In other words, f can undo or "reverse" g, but not
necessarily can be reversed by it.

Every function with a right inverse is necessarily a surjection. The proposition that every surjective
function has a right inverse is equivalent to the axiom of choice.

If f: X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from
its preimage f −1(B).

For example, in the first illustration, there is some function g such that g(C) = 4. There is also some
function f such that f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters
that f "reverses" g.

## Surjective composition: the

first function need not be
A surjective function. surjective.
Another surjective function. A non-surjective function.
(However, this one is not (This one happens to be (This one happens to be
an injection) abijection) an injection)

Surjections as epimorphisms
A function f: X → Y is surjective if and only if it is right-cancellative:[2] given any functions g,h:Y → Z,
whenever g o f = h o f, then g = h. This property is formulated in terms of functions and
their composition and can be generalized to the more general notion of the morphisms of
acategory and their composition. Right-cancellative morphisms are called epimorphisms. Specifically,
surjective functions are precisely the epimorphisms in the category of sets. The prefix epi is derived
from the greek preposition ἐπί meaning over, above, on.

Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right
inverse g of a morphism f is called asection of f. A morphism with a right inverse is called a split
epimorphism.

## Surjections as binary relations

Any function with domain X and codomain Y can be seen as a left-total and right-unique binary relation
between X and Y by identifying it with its function graph. A surjective function with domain X and
codomain Y is then a binary relation between X and Y that is right-unique and both left-total and right-
total.

## Cardinality of the domain of a surjection

The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its
codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense
of cardinal numbers. Specifically, if both X and Y are finite with the same number of elements,
then f : X → Y is surjective if and only if f is injective.

## Composition and decomposition

The composite of surjective functions is always surjective: If f and g are both surjective, then g o f is
surjective. Conversely, if f o g is surjective, then f is surjective (but g need not be). These properties
generalize from surjections in the category of sets to any epimorphisms in any category.

Any function can be decomposed into a surjection and an injection: For any function h: X → Z there
exist a surjection f:X → Y and an injection g:Y → Z such that h = g o f. To see this, define Y to be the
sets h −1(z) where z is in Z. These sets are disjoint and partition X. Thenf carries each x to the element
of Y which contains it, and g carries each element of Y to the point in Z to which h sends its points.
Then f is surjective since it is a projection map, and g is injective by definition.
Induced surjection and induced bijection
Any function induces a surjection by restricting its codomain to its range. Any surjective function
induces a bijection defined on a quotient of its domain by collapsing all arguments mapping to a given
fixed image. More precisely, every surjection f : A → B can be factored as a projection followed by a
bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence
relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f.
Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class [x]~, and
let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://ru.scribd.com/document/39919798/Injective-Function", "fetch_time": 1603280497000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-45/segments/1603107876307.21/warc/CC-MAIN-20201021093214-20201021123214-00646.warc.gz", "warc_record_offset": 534202891, "warc_record_length": 85581, "token_count": 3228, "char_count": 12313, "score": 4.09375, "int_score": 4, "crawl": "CC-MAIN-2020-45", "snapshot_type": "latest", "language": "en", "language_score": 0.9107751846313477, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00000-of-00064.parquet"}