What does it mean for a set to have an exponent?

Question

While learning about boolean algebra I came across this expression: $\{ 0,1 \}^n$. Given the context I understand that this is a set containing the values $0$ and $1$, but what the exponent here means is less clear, which leads me to my question: What does the exponent signify when the base is a set?

Answer 1

Typically $X^A$ denotes the set of all functions $A\to X$.

Now when $n\in\mathbb{N}$ then $X^n$ can be defined as $X^{\{1,2,\ldots,n\}}$.

Alternatively $X^n$ can be defined as the Cartesian product $X\times\cdots\times X$ of $n$ copies of $X$.

So in your case, for example $\{0,1\}^3$ will be the following set

$$\{(0,0,0), (0,0,1), (0,1,0), (0,1,1),$$ $$(1,0,0), (1,0,1), (1,1,0), (1,1,1)\}$$