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]]. | ||
| + | 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== | ||
| + | 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. | ||
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| + | ==See also== | ||
* [[Bijection]] | * [[Bijection]] | ||
* [[Surjection]] | * [[Surjection]] | ||
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 
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
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