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I'm not very familiar with the notation of this set. I just saw it in some slides on internet but I do not know how it is named (binary set or Boolean set maybe?), let alone a book reference. Can anyone give me a help?

Edit: Please, do not suggest $\mathbb{B}$ or "whatever you want". I am looking for the formal original name (and, possibly, its reference).

Edit 2: I am intended to use it in the digital communication systems context. Thus, I am prone to call this a "binary set", but I am not sure whether other authors call like that as well.

Rubem Pacelli
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    I don't think it has a name that would be recognized in any context. – Ethan Bolker Nov 29 '22 at 00:44
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    Possibly the most common name for it is '2'. :-) (Under the usual definition of the ordinal numbers) – Steven Stadnicki Nov 29 '22 at 00:44
  • @MarianoSuárez-Álvarez you gave me a chuckle this evening. I appreciate it. – Brevan Ellefsen Nov 29 '22 at 00:49
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    There are going to be dozens of ways to refer to the set, including but not limited to $\Bbb F_2, [2], 2, {0,1},$ and surely many more... What you call it is up to you and different names might imply something extra. For instance, with $\Bbb F_2$ this implies also the existence and usage of a multiplication and addition and allude to the fact that it forms a field. $[2]$ on the other hand might imply that whatever properties you might choose to talk about you might also be trying to show to be true for $[n]$ in general... – JMoravitz Nov 29 '22 at 00:49
  • In the end, "a rose by any other name" and what not... it's all up to you. – JMoravitz Nov 29 '22 at 00:49
  • It is sometime just called $2$ or $\mathbf 2$ or $\mathbb 2.$ – Thomas Andrews Nov 29 '22 at 00:51
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    @MarianoSuárez-Álvarez the naturals start at $0$ though and I'll fight anyone who says otherwise. – CyclotomicField Nov 29 '22 at 02:07
  • @MarianoSuárez-Álvarez Several authors will define $[2]$ as ${0,1}$ (including the authors I learned from). In the end, both conventions lead $[n]$ to being the prototypical "simplest $n$-element set" and really all that usually matters for our purposes is that it has $n$ elements in it and not specifically which $n$ elements those happened to be. As such, statements involving $[n]$ can be read with either interpretation and the statement still make sense and hold true. – JMoravitz Nov 29 '22 at 02:27
  • @MarianoSuárez-Álvarez you are so hilarious, but I need to define it properly in a article and I do not want to name it differently from the convention (if it exists) – Rubem Pacelli Nov 29 '22 at 11:20
  • @JMoravitz None of these are names. I am not asking for a symbol, I am asking for a name. In the same way that $\mathbb{R}$ is usually called "the set of the real numbers" or "real set", I am trying to find how $\mathbb{B}$ is called. – Rubem Pacelli Nov 29 '22 at 13:16
  • "The finite field of order two", "Bracket two", "Two", "Zero-One", ... Again, call it whatever you like. So long as it is clear what you are talking about it doesn't matter what it is called. – JMoravitz Nov 29 '22 at 13:24
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    Call it ${0,1}$ like most authors do, and give it a letter it you'll use it a ton. No notation is standard here besides explicitly writing it. – Brevan Ellefsen Nov 29 '22 at 14:51
  • @MarianoSuárez-Álvarez Welcome to SE.maths. We are a community intended to help each other by sharing knowledge and experiences. Naturally, it is sensible to search before asking, as I and everybody else do. Nevertheless, nothing replaces the share of personal experiences. A good argument is the lack of convergence of some points. However, one can say that there is probably a convergence about $\mathbb{N}$ being named, on average, the "set of natural numbers". I am looking for something equivalent to $\mathbb{B}$. If you do not know the answer, no problems, somebody else does. – Rubem Pacelli Nov 29 '22 at 23:39

1 Answers1

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There are a few names for the set containing two elements $\{0, 1\}$, depending on context and the intended interpretation of such a set.

If the values of the set are interpreted as false and true, then the set $\mathbb{B} = \{0, 1\}$ may be called a Boolean domain:

In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as $\{0, 1\}$, or $\mathbb{B}$. (Wikipedia)

If the values of the set are interpreted as integers modulo $2$ and a field structure is implied, then $\mathrm{GF}(2) = \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$ is the finite field of two elements or the Galois field of two elements:

$\mathrm{GF}(2)$ (also denoted $\mathbb{F}_2$, $\mathbf{Z}/2\mathbf{Z}$ or $\mathbb{Z}/2\mathbb{Z}$) is the finite field of two elements (GF is the initialism of Galois field, another name for finite fields). Notations $\mathbf{Z}_2$ and $\mathbb{Z}_2$ may be encountered although they can be confused with the notation of 2-adic integers. (Wikipedia)

In set theory, the set $\{0, 1\}$ is sometimes simply referred to as $2$, $\mathbf{2}$, or $\mathbb{2}$, following von Neumann's construction of ordinal numbers. Another notation sometimes used to refer to a generic "canonical" $2$-element set (without paying attention to the values of its elements) is $[2]$, although many authors define it as $\{1, 2\}$ instead of $\{0, 1\}$.

Whatever name and notation you choose, it is good practice to explicitly introduce them as such in writing.

L. F.
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  • The fun thing about the von Neumann numerals is that it's recursive. Like, $3={0,1,2}$, but since $2={0,1}$ that means $3={0,1,{0,1}}$, which in turn (since $1={0}$ and $0={}$) equals$${{},{{}},{{},{{}}}}.$$The intent isn't to have some deep insight into what numbers "really are", but rather to show that if you have some axioms describing how sets behave (an "axiomatic set theory"), through an encoding like this, they can describe how numbers and other objects behave. This means that axiomatic set theory can work as an axiom-based foundation of all of mathematics. – Akiva Weinberger Nov 29 '22 at 15:06
  • (which has nothing to do with the question but I thought it was interesting) – Akiva Weinberger Nov 29 '22 at 15:38
  • A really comprehensive answer. Thank you, the boolean set seems sensible, after all. – Rubem Pacelli Dec 11 '22 at 16:00