Is it correct notation to use $\mathbb{B}$ to denote $\{0,1\}$?

Question

I have seen $\mathbb{B}$ occasionally used to denote the set $\{0,1\}$, but I am not sure if this is standard practice or not.

My context is something like this:

"A solution to this problem is a vector $x \in \mathbb{Z}^n$",

when talking about a problem with $n$ integer decision variables.

My question is, can I use $x \in \mathbb{B}^n$ to talk about a binary vector? or is it better to write $x \in \{0,1\}^n$?

I don't really mind either way, it would be easier to be able to use $\mathbb{B}$, however it is not important enough to have to define it explicitly if it isn't standard practice.

I suppose I could also use $\mathbb{Z}_2$, but I am not using rings in anything else that I am writing, so it would maybe seem a bit odd.

Answer 1

While $\mathbb{B}$ is not standard notation like $\mathbb{R}$ or $\mathbb{C}$, it can of course be used to denote the Boolean set, if it is well-defined within the given context, i.e.,

"In the remainder of this discussion we denote by $\mathbb{B}$ the set $\{0,1\}$".

Then $\mathbb{B}$ is properly defined and can be used in an unambiguous way.

---

Idem for vectors of the type $\mathbb{B}^n$.

Answer 2

The question "Is $\mathbb{B}$ standard notation for the set $\{0,1\}$?" is, perhaps, a bit too broad. Typically, "standard" notation is dependent on context. For example, in some contexts, the notation $$ \mathbb{Z}_{p} $$ is used to denote the additive group of integers modulo $p$ in some contexts, while it is used to denote the $p$-adic integers in other contexts. Both of these notations are standard in the appropriate contexts.

A better question might be "Is $\mathbb{B}$ standard notation for the set $\{0,1\}$ in the context of $\underline{\hspace{5em}}$?" In a comment, Mauro ALLEGRANZA links to Wikipedia, which indicates that this notation is sometimes used to denote "a ball, a boolean domain, or the Brauer group of a field". A boolean domain is precisely what is being dealt with in the original question, so it seems reasonable to conclude that the answer my question is:

Yes, the notation $\mathbb{B} = \{0,1\}$ is standard in some contexts, including computer science and parts of algebra where decision variables are considered.

That being said, if you are writing something for broader consumption (e.g. a paper for a journal or lecture notes for a class; that is, anything which is going to be read by someone other than yourself), it cannot hurt to specifically define your notation. For example, the sentence fragment

let $\mathbb{B}$ denote the set $\{0,1\}$

could go near the introduction. $\mathbb{B}$ is an uncommon enough notation that it is unlikely that there will be any ambiguity (that is, it is unlikely that any reader will see that notation and automatically think that it is something else and be confused), so as long as you define it early on, you should be fine.

Alternatively, if you want to be absolutely sure that there is no chance of misunderstanding, you could always write $\{0,1\}$ throughout your document. It costs a couple of keystrokes and takes up a little more space on the page, but I doubt that anyone would fault you for it.