It is really difficult to ‘jump in’ the middle of any development of finite/infinite sets.

We do not know what theorems you have at hand.

Generally, we would know these for finite sets.

If

is a finite set and

then:

1) if there is an injection

then

.

1) if there is a surjection

then

.

There also a powerful theorem that applies to all sets:

*There is an injection if and only if there is a surjection *.

With those we show that no finite set bijects with a proper subset of itself.

The second definition is known as Dedekind Infinite.

If

then

is also infinite.

Do you see how you might use this to show equivalence with the first definition.