Suppose I have a vector $(a_1, .\dots, a_n)$ where each element is either $0$ or $1$.
Is it correct to write this in the following notation: $$ (a_1, .\dots, a_n), \quad a_i\in\{0,1\} \quad ? \tag 1 $$ Can I also write it as: $$ (a_1, .\dots, a_n) \in \{0,1\}^n \quad \tag 2? $$ Or something else?
Update based on the comments:
Maybe I explicit should include $\mathbb N$? So something like:
For $\mathbb N = \{1,2, \dots, n\}$ we have $$ (a_1, .\dots, a_n), \quad \forall a_i\in\{0,1\} \tag 3 $$ Or $$ (a_1, .\dots, a_n) \in \{0,1\}^n \tag 4 $$
I would use the notation $2^n$, and see every "vector" as the image of a function $$f:n\longrightarrow 2$$ $$i\longmapsto a_i $$ Where every $a_i\in 2$, and $2$ is the Von Neumann ordinal of cardinality $2$ ($2=\{0,1\}$)