# Show there exists a one to one function from N to S iff function f exists

## Homework Statement

Show that for a set S, there exists an injective function $\Phi$ :
N $\rightarrow$ S if and only if there exists an injective, but non-surjective
function f : S $\rightarrow$ S. (Sets S satisfying this condition are called
in nite sets.)

## 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 $\Phi: N \rightarrow S$ is injective then f: S $\rightarrow$ S is injective but not surjective
ii)If f: S $\rightarrow$ S is injective but not surjective then $\Phi: N \rightarrow S$ 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?

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Dick
Homework Helper
Take them one at a time. If you have phi:N->S is injective, you want to prove there is a function f:S->S that's injective but not surjective. You probably want to define f in terms of phi. Can you think of a function from N->N that's injective but not surjective?

Take them one at a time. If you have phi:N->S is injective, you want to prove there is a function f:S->S that's injective but not surjective. You probably want to define f in terms of phi. Can you think of a function from N->N that's injective but not surjective?
yes, well
n--->2n is injective but not surjective

Dick