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## Homework Statement

Show that for a set S, there exists an injective function [itex]\Phi[/itex] :

N [itex]\rightarrow[/itex] S if and only if there exists an injective, but non-surjective

function f : S [itex]\rightarrow[/itex] S. (Sets S satisfying this condition are called

innite sets.)

## Homework Equations

## The Attempt at a Solution

Since this is a if and only if (biconditional) statement.

I can prove this statement if i can prove the two conditional statements:

i) If [itex]\Phi: N \rightarrow S[/itex] is injective then f: S [itex]\rightarrow[/itex] S is injective but not surjective

ii)If f: S [itex]\rightarrow[/itex] S is injective but not surjective then [itex]\Phi: N \rightarrow S[/itex] is injective.

I realize that this is the step that i should take, but i just don't know how to prove these two statements..

Any help?