Meaning of a set in the exponent

Let $D = 2^\mathbb{N}$, i.e., D is the set of all sets of natural numbers.

What's the meaning of this definition? Intuitively, I would suggest that $D = \{1,2,4,...\}$ but the explanation "set of all sets" leads me to the guess that this is wrong.

• $n\to\{0,1\}{}{}{}{}$ Commented Aug 18, 2014 at 8:12
• $A^B$ denotes the set of all maps $B\to A$. Now set $2 = \{0,1\}$. Commented Aug 18, 2014 at 8:12
• Further to Asaf's comment: >In some parts of set theory where this notation can be confused with other types of exponentiation, you can see the notation $^BA$ used instead. Thanks for pointing this out, Asaf. To avoid any anomalies, you may indeed want to distinguish set exponentiation from, say, exponentiation on the natural numbers when it is defined only as repeated multiplication. Commented Aug 19, 2014 at 13:59
• @Mikasa If you were to re-write the statement $\begin{pmatrix} n \implies \begin{Bmatrix} 0, 1 \end{Bmatrix} \end{pmatrix}$ in English, then what would the English words say? Commented Mar 11, 2023 at 15:01
• @Mikasa There are many different notations in the world. Some people write $\begin{pmatrix} f: \mathbb{N} \implies \mathbb{R} \end{pmatrix}$ to indicate that $f$ is a function whose inputs are whole numbers such as 1, 2, 3 adn the outputs are decimal numbers such as $45.981$. However, I have never seen $\begin{pmatrix} n \implies \begin{Bmatrix} 0, 1 \end{Bmatrix} \end{pmatrix}$. MAybe $n$ is a whole number such as 1, 3, 7, etc. $\begin{pmatrix} n \implies \begin{Bmatrix} 0, 1 \end{Bmatrix} \end{pmatrix}$ looks like $\begin{pmatrix} 8 \implies \begin{Bmatrix} 0, 1 \end{Bmatrix} \end{pmatrix}$ Commented Mar 11, 2023 at 15:06

We write $A^B$ as the set of all functions $f\colon B\to A$. Namely $f$ is a function whose domain is $B$ and takes values in $A$.

In this case $A=\{0,1\}$ and $B=\Bbb N$. So this is the set of all functions from $\Bbb N$ into $\{0,1\}$. If we think about those as indicator functions then we have a natural way of thinking about $2^\Bbb N$ as the power set of $\Bbb N$, also denoted by $\mathcal P(\Bbb N)$, which is the set of all subsets of $\Bbb N$.

(In some parts of set theory where this notation can be confused with other types of exponentiation, you can see the notation ${}^BA$ used instead.)

• Just to add to this nice answer: The reason that we use the exponential notation $A^B$ to denote the set of all functions $f:B\rightarrow A$ derives from the fact that the size of this set $A^B$ is, in finite cases, indeed given by exponentiation: $|A^B| = |A|^{|B|}$. We can even take this as the definition of exponentiation, providing an understanding of exponentiation that works even for infinite cardinal numbers.
– Matt
Commented Aug 18, 2014 at 22:12
• Please keep comments on-topic. Thanks. Commented Aug 19, 2014 at 9:14
• @Matt: I think even in the absence of the other discussion your addendum is very on-topic (since this is a question about a notation after all). Commented Aug 19, 2014 at 9:25
• Comments are not for extended discussion; this conversation has been moved to chat. Commented Aug 19, 2014 at 23:43
• @martinkunev: "$A$ to the power of $B$", or "the set of functions from $B$ to $A$", depending on your context. Commented Jun 18, 2016 at 18:01

A power set $\mathcal P(S)$ of a set $S$ is sometimes denoted $2^S$. If $S$ is a finite set with $|S| = n$ elements, then the number of subsets of $S$ is $|\mathcal P(S)|=2^n$. This is the motivation for the notation $2^S$.

• You might want to include @DanielFischers' comment to give motivation to this notation ;) Commented Aug 18, 2014 at 8:13