Difference between revisions of "Injection"

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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]].
 
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]].
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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>.
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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.
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==Examples==
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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==
 
==See also==

Latest revision as of 00:01, 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 $f:A\to B$ is an injection if for all $x,y\in A$, if $f(x)=f(y)$ then $x=y$.

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

Examples

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

See also


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