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https://texercises.com/exercise/inverses/
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Exercise:
abcliste abc Let f:X rightarrow Y be a map. Ase that there exist maps g_ :Y rightarrow X and g_ : Y rightarrow X such that g_ circ f textid_X quad textand quad f circ g_ textid_Y Show then that g_ g_. What can you say about f? abc Give an example of a function that admits a right-inverse but is not bijective. By right-inverse we mean that letting f be a map from X to Y there exists a map g from Y to X such that f circ g textid_Y. abcliste

Solution:
abcliste abc We have g_ g_ circ textid_Y g_ circ f circ g_ g_ circ f circ g_ textid_X circ g_ g_. So g_ g_ f^-. Moreover since f circ g_ textid_Y f is surjective. Additionally since g_ circ f textid_X f is injective. abc Consider the maps f: mathbbR rightarrow mathbbR_geq x mapsto x^ and g: mathbbR_geq rightarrow mathbbR x mapsto sqrtx We clearly have f circ g textid_mathbbR_geq so g is a right-inverse for f. However f is not bijective as it is not injective. abcliste
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Exercise:
abcliste abc Let f:X rightarrow Y be a map. Ase that there exist maps g_ :Y rightarrow X and g_ : Y rightarrow X such that g_ circ f textid_X quad textand quad f circ g_ textid_Y Show then that g_ g_. What can you say about f? abc Give an example of a function that admits a right-inverse but is not bijective. By right-inverse we mean that letting f be a map from X to Y there exists a map g from Y to X such that f circ g textid_Y. abcliste

Solution:
abcliste abc We have g_ g_ circ textid_Y g_ circ f circ g_ g_ circ f circ g_ textid_X circ g_ g_. So g_ g_ f^-. Moreover since f circ g_ textid_Y f is surjective. Additionally since g_ circ f textid_X f is injective. abc Consider the maps f: mathbbR rightarrow mathbbR_geq x mapsto x^ and g: mathbbR_geq rightarrow mathbbR x mapsto sqrtx We clearly have f circ g textid_mathbbR_geq so g is a right-inverse for f. However f is not bijective as it is not injective. abcliste
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eth, hs22, linalg i, serie2
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