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. | ||
=Excerpt from Wikipedia= | |||
The following paragraph is copied from Wikipedia. | |||
==={{anchor|Compositional inverse}}Inverses and composition=== | |||
{{See also|Inverse element}} | |||
If {{mvar|f}} is an invertible function with domain {{mvar|X}} and codomain {{mvar|Y}}, 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>.<ref name=":2" /> | |||
Using the [[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> | |||
where {{math|id<sub>''X''</sub>}} is the [[identity function]] on the set {{mvar|X}}; that is, the function that leaves its argument unchanged. In [[category theory]], this statement is used as the definition of an inverse [[morphism]]. | |||
Considering function composition helps to understand the notation {{math|''f''<sup> −1</sup>}}. Repeatedly composing a function with itself is called [[iterated function|iteration]]. If {{mvar|f}} is applied {{mvar|n}} times, starting with the value {{mvar|x}}, then this is written as {{math|''f''<sup> ''n''</sup>(''x'')}}; so {{math|''f''<sup> 2</sup>(''x'') {{=}} ''f'' (''f'' (''x''))}}, etc. Since {{math|''f''<sup> −1</sup>(''f'' (''x'')) {{=}} ''x''}}, composing {{math|''f''<sup> −1</sup>}} and {{math|''f''<sup> ''n''</sup>}} yields {{math|''f''<sup> ''n''−1</sup>}}, "undoing" the effect of one application of {{mvar|f}}. | |||
Revision as of 14:22, 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.
Template:AnchorInverses and composition
If f is an invertible function with domain X and codomain Y, then
- , for every ; and , for every .[1]
Using the composition of functions, we can rewrite this statement as follows:
- and
where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. In category theory, this statement is used as the definition of an inverse morphism.
Considering function composition helps to understand the notation f −1. Repeatedly composing a function with itself is called iteration. If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) Template:= f (f (x)), etc. Since f −1(f (x)) Template:= x, composing f −1 and f n yields f n−1, "undoing" the effect of one application of f.
- ↑ Cite error: Invalid
<ref>tag; no text was provided for refs named:2