What is the name of the set $\left\{0,1\right\}^n$?

Question

My question is simple: What is the usual name given to the set $\left\{ 0,1 \right\}^n$ , where $n \in \mathbb{N}$ ?

Just for clarification, I using $\left\{ 0,1 \right\}^n$ to denote the set of all n-tuple with entries equal to $0$ or $1$.

Answer 1

I'd call them the binary strings of length $n$. Strings capture the order properties of a tuple and constraining their length ensures we have exactly $n$ values for each string. Strings often come with a concatenation operation which you wouldn't be using here, however comparing $10100$ to $(1,0,1,0,0)$ we see that the notation is much more readable if you're going to discuss elements of $\{0,1\}^n$ explicitly. It's for that reason I prefer to call them binary strings of length $n$.

Answer 2

I think of this as the power set of a set with $n$ elements, denoted $P(n)$. This is because what you have can be seen to have $2^n$ elements, which in turn counts the number of functions from a set with $n$ elements $[ n ] \to \{0,1\}$. This is the same thing as a subset of $[n]$ because we can associate to such a function the subset consisting precisely of those elements taking value $1$.

So I propose that the name you seek is 'power set of $n$.'