Cardinality of infinite direct sum of Z/2Z

Question

Consider the direct sum of countably many Z/2Z groups, which I'll denote by G=$⨁_n $(Z/2Z) and where the index is to keep track of each copy of Z/2Z. How is G countable? Since each Z/2Z has 2 elements,is not the cardinality 2^No?

Answer 1

You have to distinguish between the direct product and the direct sum. Informally, the elements of the direct product are arbitrary sequences of elements from $\mathbb{Z}/2\mathbb{Z}$, while the direct sum consists only of those elements in which all but finitely many terms are the identity. E.g. $$(1, 1,1, 1, 1, ...)$$ is an element of the direct product $\prod_\mathbb{N}\mathbb{Z}/2\mathbb{Z}$ but not of the direct sum $\bigoplus_\mathbb{N}\mathbb{Z}/2\mathbb{Z}$.

Since $\mathbb{Z}/2\mathbb{Z}$ only has one nonzero element, an element of the direct product or direct sum is completely determined by which of its terms are nonzero. So the question of the cardinality of $\bigoplus_\mathbb{N} \mathbb{Z}/2\mathbb{Z}$ boils down to:

How many finite subsets of $\mathbb{N}$ are there?