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| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
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inverse_function [2013/06/25 15:46] 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 $ | | + | |
| ==== Ramifications ==== | ==== Ramifications ==== | ||
| Line 12: | Line 10: | ||
| $\text{im}(f^{-1})=\text{dom}(f)=X,$ | $\text{im}(f^{-1})=\text{dom}(f)=X,$ | ||
| - | $\hspace{1cm} \text{dom}(f^{-1})=\text{im}(f).$ | + | $\text{dom}(f^{-1})=\text{im}(f).$ |
| 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: $f^{-1}\circ f=\text{id}$. | ||
