[Math] the mathematical symbol for “nothing”

foundationsnotation

The question might sound weird. But I have a situation coming up while writing a research paper. I will try to put it simply.

I want to define a random variable $X$ which takes the values from the set $\{0,1\}$. The probability of $X = 0$ is 0.4 and the probability of $X = 1$ is also 0.4. I want $X$ to take a null value with the remaining probability of $0.2$. The question is, what should be the right symbol to denote a "null value"/"nothing".

I do not want to use the symbol $\emptyset$ since it represents an empty "set".

I do not want to use any other integer say "2" or "3" or some greek symbol say "$\alpha$" or "$\tau$" since it does not reflect "nothingness" in it.

What I want to know is, Is there any well-defined notion of "nothingness" in mathematics and a corresponding symbol for it?

Best Answer

This is sufficiently rare, and handled in sufficiently many different ways, that you should always state explicitly how you're treating it. In my experience, the most common symbols are:

  • $\mathsf{null}$, $\mathsf{nil}$, $\mathsf{NaN}$ or similar on the more computer-sciencey side, and

  • $\perp$ or $\uparrow$ on the more logicy side.

    • Note that "$\perp$" is also used to denote contradiction, and "$\uparrow$" is also used as a predicate to denote "is undefined" or "doesn't halt" with "$\downarrow$" denoting "is defined"/"does halt."

But again, I'd explicitly state which you're using - although admittedly multiple of these would almost certainly make it obvious from context.

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