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A binary variable is very well used in many engineering problems.

Suppose that we need to introduce some variables that indicate whether a computer is being used by a customer or not. Then, we may express it as $x_i\in\{0,1\}$ for computer $i$. If we have 100 computers, we may use $\mathbf{x}=\{x_i\}_{i=1}^{100}$. In this case, we can write $\mathbf{x}\in\{0,1\}^{100}$ or $\mathbf{x}\in\{0,1\}^{100\times1}$.

Like an above example, $\{0,1\}$ is very well used. Is there any notation or name for $\{0,1\}$?

For example, we have $\mathbf{1}_{N\times M}$ and $\mathbf{0}_{N\times M}$ to express a set (or a matrix) with all-ones elements or all-zero elements.

Danny_Kim
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  • I do not think that there is something better than what you mentioned. – Peter Jan 19 '23 at 07:43
  • The whole $\mathbf{x}$ will be called a bit vector or binary vector but usually no special name is given to the ${0,1}$ itself. – Michal Adamaszek Jan 19 '23 at 07:50
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    Your last examples differ from ${0,1}$ by only one character. The ${0,1}$ is already very short. So whatever alias you come up with, it will either be longer or of pretty much the same size. What's the point then? On the other hand ${0,1}$ is straight forward and easy to understand. – freakish Jan 19 '23 at 07:52
  • Let me give you another example. Whenever I use $S^n$ I always explain at least once what it means. But that's because the definition of $S^n={v\in\mathbb{R}^{n+1}\ |\ \lVert v\rVert=1}$ is a lot longer. – freakish Jan 19 '23 at 07:56
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    In Set Theory the usual name for ${0,1}$ is $2$. [In the same way, the name of ${0}$ is $1$.] – ancient mathematician Jan 19 '23 at 07:57
  • Thant you all. Thanks to your comments, I knew that there are no speical names for ${0,1}$. :-) – Danny_Kim Jan 19 '23 at 12:37

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As far as I know, in mathematics it's usually just called $\{ 0, 1 \}$. Sometimes it's identified with the set of truth values but as far as I know there isn't standard notation for that either. One could contemplate using $\mathbb{B}$ for "Boolean" but this is not at all standard.

$\{ 0, 1 \}$ as a set is an example of a type of object called a subobject classifier in category theory, the standard notation for which is $\Omega$. But that's not at all a standard way to refer to $\{ 0, 1 \}$ either.

Finally, in some contexts it is just referred to as $2$, but that's likely to be confusing if you don't explicitly note it.

Qiaochu Yuan
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