Cardinality of all the functions from $\mathbb N$ to $\{0,1\}$.

Question

Is it true to say that: $$|\{0,1\}^\mathbb N| = |\{0,1\}|^{|\mathbb N|} = 2^{\aleph_0}=\aleph$$

As I know the right part of the equation is true, but I don't know if the equations to it are allowed.

Answer 1

Yes, that is valid cardinal arithmetic. In fact, this nice algebraic manipulation was the reason why the "power set" notation exists.

The number of functions $f : \mathbb{N} \to \{0, 1\}$ is $\{0, 1\}^\mathbb{N}$, which is also the number of subsets of $\mathbb{N}$ (if $n$ is inside the subset then $f(n) = 1$, otherwise $f(n) = 0$), hence the power set cardinality $2^{|\mathbb{N}|}$.

It is worth noting that $2$, defined as an ordinal, is equal to $\{0, 1\}$. Nonnegative integers can be seen as the set of all nonnegative integers smaller than them. Hence, $\{0, 1\}^\mathbb{N} = 2^\mathbb{N}$!

Answer 2

Yes. It is true, and it's fine to do that.

Recall the basics of cardinal exponentiation: $$\left|A^B\right|=|A|^{|B|}$$

Since $|\{0,1\}|=2$ and $|\mathbb N|=\aleph_0$, the equation holds.