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Let X be a non-empty set and f a mapping of X into itself. Show that f is one-to-one onto iff there exists a mapping g of X into itself such that fg = gf = iX. If there exists a mapping g with this property, then there is only one such mapping. Why?

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Topology
Sets and Functions (XXXIX)
Functions

Let X be a non-empty set and f a mapping of X into itself.
Show that f is one-to-one onto iff there exists a mapping g of X into itself
such that fg = gf = iX.
If there exists a mapping g with this property, then there is only one such mapping. Why?

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Topology
Sets and Functions (XXXIX)
Functions ...

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This solution is comprised of a detailed explanation of the properties of the mappings.
It contains step-by-step explanation of the following problem:

Let X be a non-empty set and f a mapping of X into itself.
Show that f is one-to-one onto iff there exists a mapping g of X into itself
such that fg = gf = iX.
If there exists a mapping g with this property, then there is only one such mapping. Why?

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