countable or not countable

Question

Good evening everyone;

Can you tell me whether are they countable or not ? $$ 2^\mathbb{R}\\ 2^\mathbb{Z}\\ 2^\mathbb{N} $$ where $\mathbb{R}$ is the set of real numbers. $\mathbb{Z}$ is set of integers. $\mathbb{N}$ is the set of natural numbers.

In my opinion, all of them are uncountable. However, I would like to hear your opinion.

Regards,

Answer 1

I will just give you a hint: No, the sets are not countable. For the $2^\mathbb{N}$ case start by assuming that it is countable. Then take a bijection $f: \mathbb{N} \to 2^\mathbb{N}$. Then consider the set $$ X = \{n\in \mathbb{N} : n\notin f(n)\}. $$ Now see if you can get a contradiction.

Answer 2

Hint: suppose $2^\mathbb N$ is countable. Then, construct a sequence whose $nth$ term is different from the $nth$ term of the $nth$ element of $2^ \mathbb N$. What's unusual about such a construction?

Answer 3

HINT: Recall that $|X|<|2^X|$, and that countable sets are those sets $A$ such that $|A|\leq|\Bbb N|$.