1. are these sets countable

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{0,1,2,...,9}^N

N is the set of natural number

2. Originally Posted by szpengchao
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Not this is $2^{\aleph_0}$.

{0,1,2,...,9}^N
$10 = {0,1,...,9}$ and $2\leq 10$ thus $|2^{\mathbb{N}} | \leq |10^{\mathbb{N}}|$.
Thus, it cannot be countable.