In a binary vector $ x \in \{0,1\}^{k}$ what does the $^{k}$ mean?

Question

In a binary vector $ x \in \{0,1\}^{k}$ what does the $^{k}$ mean? I understand that $\in$ means 'is a possible outcome' or 'in' so x can be 0 or 1, but I'm not sure what the $^{k}$ means.

Answer 1

Just as points in $\mathbb R^k$ consist of ordered tuples of $k$ real numbers, so points in $\left\{0,1\right\}^k$ consist of ordered tuples of $k$ "bits".

The typical point $x = (x_1,x_2,\ldots,x_k)$, where each $x_i\in\left\{0,1\right\}$ (that is, each $x_i$ is $0$ or $1$).