Question

Use the Cantor-Schr oder-Bernstein theorem to prove that (0, ∞) and R have the same car- dinality. (Hint: Use an exponential function).

Answer 1

Let, f : R----> (0, ∞) be defined by, f(x) = e^x for all x in R

Then, f(x) > 0 for all x, i.e. f(x) belongs to (0,∞)

Injective :

Let, f(x) = f(y)

So, e^x = e^y

So, x = y (since, e^x is monotonically increasing on R)

So, f(x) = f(y) implies x = y

Hence, f is injective.

Surjective :

Let, y belongs to (0,∞)

Then, y > 0, so, ln(y) is defined.

Now, f(ln(y)) = e^ln(y) = y

So, f is onto.

Hence, f is a bijection from R onto (0,∞)

So, R & (0,∞) are equipotent & hence, have the same cardinality.