Difference between revisions of "Inverse"
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: <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 <math>\operatorname{id}_X</math> is the [[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 [[morphism]]. | where <math>\operatorname{id}_X</math> is the [[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 [[morphism]]. | ||
Revision as of 14:26, 6 August 2021
The notion of inverse in mathematical operation is simply the anti-operator.
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.
