Difference between revisions of "Bijection"
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A bijection is a 1:1 map. A map which is both [[Injection|injective]] and [[Surjection|surjective]]. | A bijection is a 1:1 map. A map which is both [[Injection|injective]] and [[Surjection|surjective]]. | ||
It has the useful property that for <math>f:X\rightarrow Y</math> that <math>f^{-1}(y)</math> is always defined, and is at most one element. | It has the useful property that for <math>f:X\rightarrow Y</math> that <math>f^{-1}(y)</math> is always defined, and is at most one element. | ||
| − | Thus <math>f^{-1}</math> behaves as a normal function (rather than the always-valid but less useful <math>f^{-1}:Y\rightarrow\mathcal{P}(X)</math> where <math>\mathcal{P}(X)</math> denotes the [[Power | + | Thus <math>f^{-1}</math> behaves as a normal function (rather than the always-valid but less useful <math>f^{-1}:Y\rightarrow\mathcal{P}(X)</math> where <math>\mathcal{P}(X)</math> denotes the [[Power set|power set]] of <math>X</math>) |
| − | + | ==Statements to add (to new version of the page, 'cos current version is dire)== | |
| − | {{Definition}} | + | * [[Every bijection yields an inverse function]] |
| + | {{Definition|Set Theory|Elementary Set Theory}} | ||
Latest revision as of 11:49, 26 September 2016
This page is a dire page and is in desperate need of an update.
The message is:
This must have been one of the first pages, from Feb 2015!
A bijection is a 1:1 map. A map which is both injective and surjective.
It has the useful property that for [math]f:X\rightarrow Y[/math] that [math]f^{-1}(y)[/math] is always defined, and is at most one element.
Thus [math]f^{-1}[/math] behaves as a normal function (rather than the always-valid but less useful [math]f^{-1}:Y\rightarrow\mathcal{P}(X)[/math] where [math]\mathcal{P}(X)[/math] denotes the power set of [math]X[/math])
