Suppose X and Y are countably infinite sets Show that X and

Suppose X and Y are countably infinite sets. Show that X and Y is also countable

Solution

Suppose X and Y are countably infinite sets.  The elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number.

A set S is countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}.^[5]

If such an f can be found that is also surjective (and therefore bijective), then S is called countably infinite.

In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N.

Hence, X and Y are countable.