Medida (matemÃ¡ticas)

A function $f: X \ rightarrow Y$ is known as surjective or is a surjection so for all y in the Ensemble of arrival Y, it exists at least an element x of the X source such as f (x) = y. It is said whereas any element y Y admits at least a previous x (by f).

In an equivalent way, it is said that $f$ is surjective if the direct image $f \ star (D_f)$ is equal to the whole of arrival Y, with Df the Ensemble of definition of F.

When X and Y are all the two equal ones to the real right $\ mathbb R$ , then a surjective function $f: \ mathbb R \ rightarrow \ mathbb R$ has a graph which intersects any horizontal line.

If a surjection is also a injection, then it is called a Bijection.

Formal definition

That is to say

f

an application of

E

F

f

is known as surjective if and only if

$\ forall there \ in F, \, \ exist X \ in E, \, F (X) =y$

Concrete example

Let us take the case of a station of holidays when a group of tourists must be placed in a hotel. Each way of dividing these tourists in the hotel rooms can be represented by an application of the whole of the tourists towards the whole of the rooms (with each tourist a room is associated).

the tourists wish that the application be injective , i.e. each one of them has an individual room . That is not possible that if the number of tourists does not exceed the number of rooms.
the hotel one wishes that the application be surjective , i.e. each room is occupied . That is not possible that if there are at least as many tourists as of rooms.
These desideratas are compatible only if the number of tourists is equal to the number of rooms. In this case, it will be possible to divide the tourists so that there is only one by room of them, and that all the rooms are occupied: the application will be then at the same time injective and surjective; it will be said that it is bijective .

Examples and counterexamples

Functions on realities:

f: \ mathbb R \ rightarrow \ mathbb R

F (X) = X ²

is not surjective, because there is no X such as F (X) = -4, for example. On the other hand, if one changes the definition of F by giving his whole of arrival as being R+, then it becomes it.

Let us consider the function $f: \ mathbb R \ rightarrow \ mathbb R$ defined by f (x) = 2x + 1. This function is surjective, since for all real arbitrary y, we can find solutions of the equation y = 2x + 1 of unknown factor x; a solution is x = (y − 1) /2.

On the other hand, the function $g: \ mathbb R \ rightarrow \ mathbb R$ defined by g (x) = (cos x) ² is not not surjective, because (for example) there does not exist reality x such as (cos x) ² = - 1.

In addition, if we define the function $h: \ mathbb R \ rightarrow \ mathbb$ consequently relation that g, but with a Together of arrival which was restricted with all the real numbers ranging between 0 and 1, then the function h is surjective. That is explained by the fact why, for any arbitrary reality y of interval 1, we can find solutions of the equation y = (cos x) ² of unknown factor x which is for example x = Arccos (√y) or x = Arccos (−√y).

Properties

a function f: X → Y is surjective if and only if there exists a function g: Y → X such as f ∘ g is equal to the Application identity on Y. (this proposal is equivalent to the Axiome of the choice.)
a function is bijective if and only if it is at the same time surjective and injective.
If f ∘ g is surjective, then f is surjective.

if f and g are all the two surjective ones, then f ∘ g is surjective.

f: X → Y is surjective if and only if, for all functions given g, h: Y → Z, when g ∘ f = h ∘ f, then g = h. In other words, the surjective functions are precisely the epimorphism S in the categories of units.
If f: X → Y is surjective and B is a Sous-ensemble of Y, then f (f⁻¹ (B)) = B. Thus, B can find starting from the direct Image of f⁻¹ (B).

Any function h: X → Z can be broken up like h = g ∘ f for a suitable surjection f and injection g. This decomposition is single near with a isomorphism, and f can be considered function taking same the values as h but with its whole of arrival restricted to the image of h, h (X), which is only one subset of the whole of arrival Z of h.
If f: X → Y is a surjective function, then X has at least as many elements as Y, within the meaning of the cardinal . (this proposal is also equivalent to the axiom of the choice.)

See too

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