Prove that if A is any countably infinite set, B is any set, and g : A → B is onto, then B is countable.

To prove:

If A is any countably infinite set, B is any set and g:A→B is onto, then B is countable.

Given information:

A is any countably infinite set, B is any set and g:A→B is onto.

Proof:

Given:

A is a countable infinite set.

B is a set.

g:A→B is onto.

Let us define the function f as:

f:B→A,f(x)=y for some element y such that x=g(y)

f one-to-one:

Let a∈B and b∈B such that a andb have the same image:

f(a)=f(b)

However, we know that the image of a is an element y such that a=g(y).

Similarly, we know that the image of b is an element z such that b=g(z).

f(a)=y⇔a=g(y)

f(b)=z⇔b=g(z)

f(a)=f(b) then implies that the two elements y and z need to be the same

y=