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an infinite set S to itself is one-to-one if and only if it is onto [closed]

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prove that a function from a finite set S to itself is one-to-one if and onlyif it is onto. Is this true when S is infinite?

the foregoing question is from contemporary abst algebra by joseph gallian page $113$ question $12$.

I proved the first part which is for finite sets.

However,i could not do anything for the infinite case,can you help me for infinite case ? I think about some examples such as $f:R \rightarrow R$ such that $f(x)=x$,i.e, the identity.

here f is one-to-one if and onlyif it is onto

It is just example, but i want to see it using proof techniques in order to not missing counter examples.

can you please help me for its proof for infinite sets ? do you have any proof of counter example to disprove it ?

why did you downvote ?


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