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inverse_function [2013/06/25 15:45] nikolaj |
inverse_function [2014/03/21 11:11] (current) |
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===== Inverse function ===== | ===== Inverse function ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $ f\in X^Y_\text{inj} $ | | + | | @#55CCEE: context | @#55CCEE: $ f\in X^Y_\text{inj} $ | |
- | | @#55EE55: $ f^{-1} $ | | + | | @#55EE55: postulate | @#55EE55: $ f^{-1} \equiv f^\smile $ | |
- | | @#55EE55: $ f^\smile $ | | + | ==== Discussion ==== |
+ | We have | ||
- | ==== Ramifications ==== | + | $\text{im}(f^{-1})=\text{dom}(f)=X,$ |
- | We have $\text{im}(f^{-1})=\text{dom}(f)=X$ as well as $\text{dom}(f^{-1})=\text{im}(f)$. | + | |
+ | $\text{dom}(f^{-1})=\text{im}(f).$ | ||
Injectiveness of $f$ implies there is a left "left inverse" of the function: $f^{-1}\circ f=\text{id}$. | Injectiveness of $f$ implies there is a left "left inverse" of the function: $f^{-1}\circ f=\text{id}$. | ||
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Mizar files: [[http://mizar.org/JFM/Vol1/funct_1.html|FUNCT_1]] | Mizar files: [[http://mizar.org/JFM/Vol1/funct_1.html|FUNCT_1]] | ||
- | ==== Context ==== | + | ==== Parents ==== |
- | Set constructor | + | === Context === |
- | === Parents === | + | |
[[Injective function]], [[Reversed relation]] | [[Injective function]], [[Reversed relation]] |