Do you understand the distinction between countable and uncountable?

Every subset of a countable set is countable.

Every superset of an uncountable set is uncountable.

The set of integers,, is countable.Z

The set of reals,, is uncountable.R

Given those facts, answer #1 & #4.

If define the image function .

Now show that if is bijective then is bijective.

That proves #2.

For #3 note that is countable and every quadratic equation has at most two real roots.