# Difference between revisions of "Injection"

(6 intermediate revisions by 4 users not shown) | |||

Line 1: | Line 1: | ||

− | An '''injection''', or "one-to-one function," is a [[function]] takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the [[range]] is the image of exactly one value in the [[domain]]. | + | An '''injection''', or "one-to-one function," is a [[function]] that takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the [[range]] is the image of exactly one value in the [[domain]]. |

− | ==See also== | + | Alternative definition: A function <math>f:A\to B</math> is an injection if for all <math>x,y\in A</math>, if <math>f(x)=f(y)</math> then <math>x=y</math>. |

+ | |||

+ | The binary relation <math>|X|\leq|Y|</math> iff there is an injection <math>f:X\rightarrow Y</math> forms a partial order on the class of cardinals: <math>X\leq X</math>, <math>X\leq Y</math> and <math>Y\leq X</math> implies <math>|X|=|Y|</math> by the Cantor-Schroeder-Bernstein theorem, and <math>|X|\leq|Y|</math> and <math>|Y|\leq|Z|</math> implies <math>|X|\leq|Z|</math> because the composition of injections is again an injection. | ||

+ | |||

+ | ==Examples== | ||

+ | Linear functions are injections: <math>f:\mathbb R \to \mathbb R</math>, <math>f(x)= ax+b</math>, <math>a\neq 0</math>. The domain choosing is also important. For example, while <math>f:\mathbb R \to \mathbb R</math>, <math>f(x)=x^2</math> is not an injection (<math>f(-1)=f(1)=1</math>), the function <math>g:[0,\infty)\to\mathbb R</math>, <math>g(x)=x^2</math>, is an injection. | ||

+ | |||

+ | ==See also== | ||

* [[Bijection]] | * [[Bijection]] | ||

* [[Surjection]] | * [[Surjection]] | ||

+ | |||

+ | |||

+ | {{stub}} |

## Latest revision as of 01:01, 17 November 2019

An **injection**, or "one-to-one function," is a function that takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the range is the image of exactly one value in the domain.

Alternative definition: A function is an injection if for all , if then .

The binary relation iff there is an injection forms a partial order on the class of cardinals: , and implies by the Cantor-Schroeder-Bernstein theorem, and and implies because the composition of injections is again an injection.

## Examples

Linear functions are injections: , , . The domain choosing is also important. For example, while , is not an injection (), the function , , is an injection.

## See also

*This article is a stub. Help us out by expanding it.*