How to represent set of numbers defined by $b$ bits?

Question

I want a concise representation for the set of numbers defined by $b$ bits. For example, I want to write $x \in \mathbb{R}^{n}$ except instead of $\mathbb{R}$ I want all number between $0$ and $2^b-1$ where $b$ is the number of bits.

I have no idea what tags to use for this question. Sorry.

Answer 1

[Converted to answer from comment]

Why not just use $\{0,1\}^b = \{0,1\}\times\cdots\times \{0,1\}$, the Cartesian product of a 2-element set with itself $b$ times? The elements of this set correspond exactly with the set of numbers you mention.

Answer 2

For the numbers $n$ with no more than $b$ bits, the valid values are $0 \leq n \leq 2^b-1$.

For the numbers $m$ with exactly $b$ bits (highest bit is $1$) then the values are $2^{b-1} \leq m \leq 2^b-1$.

The number of binary bits is independent of the base you express the number in, so defining it over the natural numbers is fine.