# 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?

See the attached file.

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##### Solution Summary

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?

##### Solution Preview

Topology

Sets and Functions (XXXIX)

Functions ...

###### Education

- BSc, Manipur University
- MSc, Kanpur University

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