Difference between revisions of "Inverse"
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The notion of [[wikipedia:Inverse function|inverse]] in mathematical operation is simply the anti-operator. | The notion of [[wikipedia:Inverse function|inverse]] in mathematical operation is simply the [[anti-operator]]. Note that it is different from [[reverse]], where it can just be used to mean the order of presentation being reversed. | ||
=Excerpt from Wikipedia= | =Excerpt from Wikipedia= | ||
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===Compositional inverseInverses and composition=== | ===Compositional inverseInverses and composition=== | ||
If | If <math>f</math> is an invertible function with domain <math>X</math> and codomain <math>Y</math>, then | ||
: <math> f^{-1}\left( \, f(x) \, \right) = x</math>, for every <math>x \in X</math>; and <math> f\left( \, f^{-1}(y) \, \right) = y</math>, for every <math>y \in Y. </math>. | : <math> f^{-1}\left( \, f(x) \, \right) = x</math>, for every <math>x \in X</math>; and <math> f\left( \, f^{-1}(y) \, \right) = y</math>, for every <math>y \in Y. </math>. | ||
Using the [[composition of functions]], we can rewrite this statement as follows: | Using the [[wikipedia:Function composition|composition of functions]], we can rewrite this statement as follows: | ||
: <math> f^{-1} \circ f = \operatorname{id}_X</math> and <math>f \circ f^{-1} = \operatorname{id}_Y, </math> | : <math> f^{-1} \circ f = \operatorname{id}_X</math> and <math>f \circ f^{-1} = \operatorname{id}_Y, </math> | ||
where { | where <math>\operatorname{id}_X</math> is the [[wikipedia:Identity function|identity function]] on the set <math>X</math>; that is, the function that leaves its argument unchanged. In [[Category Theory]], this statement is used as the definition of an inverse [[wikipedia:morphism|morphism]]. | ||
==Related Pages== | |||
*[[Definition::Anti-operator]] | |||
Latest revision as of 08:30, 19 August 2021
The notion of inverse in mathematical operation is simply the anti-operator. Note that it is different from reverse, where it can just be used to mean the order of presentation being reversed.
Excerpt from Wikipedia
The following paragraph is copied from Wikipedia.
Compositional inverseInverses and composition
If is an invertible function with domain and codomain , then
- , for every ; and , for every .
Using the composition of functions, we can rewrite this statement as follows:
- and
where is the identity function on the set ; that is, the function that leaves its argument unchanged. In Category Theory, this statement is used as the definition of an inverse morphism.
