Show that there exists a mapping from a set S to itself that is 1-1, but not onto IFF there exists a mapping from a set S to itself that is onto, but not 1-1.(adsbygoogle = window.adsbygoogle || []).push({});

Firstly i show it, assuming that the one-one mapping (but not onto) exists.

Now i know that if there is a funtion that is 1-1, but not onto, to define a mapping f on the set (call it S), to a subset of S (call it T). This is such that f: S -> T is one-one and onto. Next, I define a mapping g:S->S such that

g(x) = { f^-1(x), when x is an element of T

{ x, whenever x is an element of S\T

That is my mapping that is then onto, but not one-one.

However, I'm stuch on the converse. That is, how do i prove, assuming a mapping from S to itself that is onto, but not one-one, implies that there is a mapping that is one-one, but not onto. ANY HELP>?

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# Proof regarding 1-1, and onto mappings

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