What is the mathematical symbol for "nothing"?

Question

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?

Answer 1

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.

Answer 2

If you are thinking of 0 and 1 as strings, there is the null (empty) string $\lambda.$

By definition, if you concatenate $$\lambda x=x\lambda=x,$$ for any string $x$ in any alphabet.

Answer 3

You can use some kind of Many-valued logic, but you said you want to put it simply. In SQL there is 3-valued logic with "null"/"unknown", for example.