What does it mean $2^\mathbb{N}$?

Question

I have an exercise that involves this set: $2^\mathbb{N}$, but I don't know what set it is. I know that the elements of the set are functions whose domain is $\mathbb{N}$, but no idea which type of functions they are.

Thanks in advance.

Answer 1

$2^\mathbb{N}$ can mean the power set of $\mathbb{N},$ which is the set of all subsets of $\mathbb{N}.$

$2^\mathbb{N}$ can also mean the set of all functions from $\mathbb{N}$ to $\lbrace 0, 1 \rbrace.$ (In set theory, $2$ is taken to mean the set $\lbrace 0, 1 \rbrace.)$

These two sets have a canonical bijection between them.

Answer 2

$2^\Bbb N$ is another notation for $\mathcal P(\Bbb N)$ which is the power set of the natural numbers; the set of all subsets of the natural numbers.

To generate a subset you decide which elements of the set to include.   This decision is basically a function mapping the set to $\{0, 1\}$ (don't or do include).

Thus the set of subsets is bijective with a set of all such functions, and $2^\Bbb N$ can also be used as a notation for this.

$$2^\Bbb N \equiv \{ f\mid f:\Bbb N\mapsto \{0,1\}\}$$

Which leads to similar notations such as:

$$3^\Bbb N \equiv \{ f\mid f:\Bbb N\mapsto \{0,1,2\}\}$$

Answer 3

In set theory, if $A$ and $B$ are sets, $A^B$ (sometimes written $^BA$) is the set of all functions $f:B\to A.$ To interpret $2^\mathbb N$ you have to know what sets $\mathbb N$ and $2$ are. $\mathbb N$ is either the set of all positive integers or the set of all nonnegative integers, depending on what book you're using. If $2$ is a set, it's probably the set $\{0,1\}.$ So $2^\mathbb N$ is the set of all infinite sequences of zeros and ones.