Surjection

Surjective function.

Surjection (from fr. Sur “on, above” + lat. Jacio “throw”), surjective mapping - the mapping of the set $X$ ${\ displaystyle X}$ $X$ to many $Y$ ${\ displaystyle Y}$ $Y$ $(f\colon X\to Y)$ ${\ displaystyle (f \ colon X \ to Y)}$ ${\ displaystyle (f \ colon X \ to Y)}$ at which each element of the set $Y$ ${\ displaystyle Y}$ $Y$ is an image of at least one element of the set $X$ ${\ displaystyle X}$ $X$ , i.e $\forall y\in Y\exists x\in X:y=f(x)$ ${\ displaystyle \ forall y \ in Y \ exists x \ in X: y = f (x)}$ $\ forall y \ in Y \ exists x \ in X: y = f (x)$ , in other words, a function that takes all possible values. It is sometimes said that surjective mapping $f\colon X\to Y$ ${\ displaystyle f \ colon X \ to Y}$ $f \ colon X \ to Y$ displays $X$ ${\ displaystyle X}$ $X$ on $Y$ ${\ displaystyle Y}$ $Y$ (as opposed to the injective mapping , which displays $X$ ${\ displaystyle X}$ $X$ at $Y$ ${\ displaystyle Y}$ $Y$ )

The concept of surjection (along with injection and bijection ) was introduced into the works of Bourbaki and became widespread in almost all branches of mathematics.

Properties

Display $f\colon X\to Y$ ${\ displaystyle f \ colon X \ to Y}$ $f\colon X\to Y$ surjective if and only if the image of the set $X$ ${\ displaystyle X}$ $X$ when displaying $f$ ${\ displaystyle f}$ $f$ coincides with $Y$ ${\ displaystyle Y}$ $Y$ : $f(X)=Y$ ${\ displaystyle f (X) = Y}$ $f(X)=Y$ . Also surjectivity of the function $f$ ${\ displaystyle f}$ $f$ equivalent to the existence of a right inverse mapping , i.e. such a mapping $g\colon Y\to X$ ${\ displaystyle g \ colon Y \ to X}$ $g\colon Y\to X$ , what $f(g(y))=y$ ${\ displaystyle f (g (y)) = y}$ $f(g(y))=y$ for anyone $y\in Y$ ${\ displaystyle y \ in Y}$ $y\in Y$ (in functional notation - $f\circ g=\mathbf {Id} _{Y}$ ${\ displaystyle f \ circ g = \ mathbf {Id} _ {Y}}$ $f\circ g=\mathbf {Id} _{Y}$ )

Examples

$f\colon \mathbb {R} \to [-1;\;1],\;f(x)=\sin x$ ${\ displaystyle f \ colon \ mathbb {R} \ to [-1; \; 1], \; f (x) = \ sin x}$ $f\colon \mathbb {R} \to [-1;\;1],\;f(x)=\sin x$ - surjectively.
$f\colon \mathbb {R} \to \mathbb {R} _{+},\;f(x)=x^{2}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R} _ {+}, \; f (x) = x ^ {2}}$ $f\colon \mathbb {R} \to \mathbb {R} _{+},\;f(x)=x^{2}$ - surjectively.
$f\colon \mathbb {R} \to \mathbb {R} ,\;f(x)=x^{2}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}, \; f (x) = x ^ {2}}$ $f\colon \mathbb {R} \to \mathbb {R} ,\;f(x)=x^{2}$ - is not surjective (for example, there is no such $x\in \mathbb {R}$ ${\ displaystyle x \ in \ mathbb {R}}$ $x\in \mathbb {R}$ , what $f(x)=-9$ ${\ displaystyle f (x) = - 9}$ $f(x)=-9$ )

Usage

In topology, the important concept of bundle is defined as an arbitrary continuous surjective mapping of topological spaces (bundle space into the base of the bundle).

The organization of a many-to-one relationship between tables in the entities of a relational data model can also be considered as a surjective function.

In category theory, the concept of surjection is generalized to the concept of epimorphism , moreover, in many categories these concepts coincide, but in the general case this is not so.

Literature

N.K. Vereshchagin, A. Shen. Beginnings of set theory // Lectures on mathematical logic and theory of algorithms . (inaccessible link)
Ershov Yu. L., Palyutin EA. Mathematical logic: textbook. - 3rd, stereotype. ed. - SPb. : Doe, 2004 .-- 336 p.

Surjection

Properties

Examples

Usage

Literature

See also

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