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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==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==

Revision as of 14:48, 25 September 2007

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 $x=y$.

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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