TITLE: What's the meaning of a set to the power of another set?
QUESTION [20 upvotes]: ${ \mathbb{N} }^{ \left\{ 0,1 \right\}  }$ and ${ \left\{ 0,1 \right\}  }^{ \mathbb{N} }$ to be more specific, and is there a countable subset in each one of them? How do I find them?

REPLY [5 votes]: Indeed, we can identify a function $f:Y\to X$ with an element of $\prod_Y X$ (a direct product of copies of $X$ indexed by $Y$) by considering the component with index $y$ the image of $y$ under $f$. This lends to what I feel is an intuitive way to remember the meaning of $X^Y$ with very little effort.
For whole numbers $n$, we have $a^n=\underbrace{a\times a\times\cdots\times a}_n$. Similarly for sets we may choose to write
$$X^Y\cong \underbrace{X\times X\times \cdots \times X}_Y.$$
All the better if one is already used to thinking of indexed tuples as functions of the index set.