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inverse_function [2013/06/25 15:45] nikolaj |
inverse_function [2013/08/21 11:59] nikolaj |
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| | @#88DDEE: $ f\in X^Y_\text{inj} $ | | | @#88DDEE: $ f\in X^Y_\text{inj} $ | | ||
| - | | @#55EE55: $ f^{-1} $ | | + | | @#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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| ==== Context ==== | ==== Context ==== | ||
| - | Set constructor | + | === Requirements === |
| - | === Parents === | + | |
| [[Injective function]], [[Reversed relation]] | [[Injective function]], [[Reversed relation]] | ||
