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1. If the set $S$ is countably infinite, prove or disprove that if $f$ maps $S$ onto $S$ (i.e $f:S \rightarrow S$ is a surjective function), then $f$ is one-to-one.

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this statement always doesn't hold.It depends on what the function f is.

Suppose the mapping is from N(natural number set,countably infinite) -> N.And  f(i)=  $\left \lceil \frac{i}{2} \right \rceil$.it is onto function but not one-one.

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