Prove that if A and B are uncountable then Ax B is uncountab

Prove that if A and B are uncountable then Ax B is uncountable.

Solution

AXB is uncountable if one of the sets is uncountable i.e, if A is uncountable, then AXB is uncountable even if B is countable. It can be showed as following. Here Let X be an uncountable set, and let Y be any set uncountable or countable or any set but it is is non-empty).

Let a belongs X and b belongs Y. Then the set

G = X x Y = {g = (a x b)}

Suppose that G is an infinite, but countable set. then we can set up a bijection f from the natural numbers N = {1, 2, 3,............} to G

Then, for all g belongs G there exists elements n ? N such that
f(n) = g
But g = a x b, which is a product of some element b belongs Y with a member of an uncountable set.
In other words there exists an element h belongs G which cannot be expressed as
f(n) = h
Therefore there cannot be a one-to-one correspondence between N and G.

Hence, the cardinality of the set of the Cartesian product G = X x Y is uncountable.