Question

1) Let W ⊂ Z+, W = {1,4,9,25,36,...}. Is set W countable?

Answer 1

Yes

Explanation:

Let's remember the definition of countable:

A countable set is a set with the same number of elements as some subset of the set of natural numbers. Whether finite or infinite, 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 unique natural number.

Let's take a look to our set W = {1,4,9,25, 36, …}.

We can rewrite it as {
`1^2, 2^2, 3^2, 5^2, 6^2, ...`}

We can associate each element of the set with a unique natural number (the number 4 it's not in the list but that's not a problem, it fulfills the definition anyway). In other words, there is a bijection between W and the natural numbers.

Therefore, W is countable.