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I think it's simpler if I write down where I wanted to use this symbol, rather than trying to explain it in an abstract way. I'll give a simple example and then explain the actual problem I was working on.

Simple example: Let $a \in \mathbb{N}$ and $b \in \mathbb{Z}$, then $a * b \in \mathbb{Z}$. The symbol I'm looking for (let's say it's $\{\cdot\}$) would be used like this: $\{\mathbb{N}\} * \{\mathbb{Z}\} = \{\mathbb{Z}\}$

The example I was working on was to prove that the Hessian matrix, $H$, of a likelihood function for a normal distribution is negative definite at the solutions to the likelihood equations. Calculating $z^{T}Hz$ where $z = (a, b)^T$ gives:

$$z^THz = \frac{-a^2n^2}{S_{xx}}+\frac{-b^2n^3}{2S_{xx}^2}$$

I know that

  • $a, b \in \mathbb{R}$ and $(a, b) \ne (0, 0)$
  • $n \in \mathbb{N^+}$
  • $S_{xx} \in \mathbb{R_{\ge 0}}$

Then it can be shown that from the two elements that are added together only one can be zero under these conditions and the other one must be negative, so the Hessian is negative definite. Is there a symbol that I could use to show something like $-\{\mathbb{R}\}^2\{\mathbb{N}\}^2 = \{\mathbb{R^-}\}$, etc. deriving the result?

The question is not about how to calculate the definiteness but if there is a symbol that could be used here?

Thanks,

Norbert

Asaf Karagila
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1 Answers1

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The closest one would be applying an operand to a set:

Let $M,N$ be two sets and $\circ$ a mathematical operation on elements of $M$ and $N$ (e.g. addition, multiplication,...). Then $$M \circ N := \{m \circ n \mid m \in M, n \in N\}.$$ So for example, $-\mathbb{N} = \{-n \mid n \in \mathbb{N}\}$ or $\mathbb{N} \cdot\mathbb{Z} = \mathbb{Z}$ or maybe the most prominent example: $V + W = \{ v + w \mid v \in V, w \in W\}$ for two vector spaces $V,W$.

In some cases, these operands on sets are commonly used, in other cases they are rather rare and it would be better to define or at least mention them first. You should also ask yourself if you really need this notation, so if a definition of it is really justified.

Dirk
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  • Thanks Dirk. I don't really need this notation, I was just curious if one exists. I was thinking of just simply using the symbols for the sets, but it gets confusing when I raise a set to a power, e.g. $\mathbb{R^2}$ is a set for vectors of dimension 2 and in my example, I would need to say something like squaring any number from $\mathbb{R}$ gives a result in $\mathbb{R}$, kind of like $f(x) = x^2, f: \mathbb{R} \to \mathbb{R}$ without defining $f$ –  Apr 08 '19 at 12:47
  • @norbertk the ambiguity of squaring is a pain. You could probably write $\mathbb R*\mathbb R\subseteq\mathbb R$ or similar, but definitely make sure you announce that you're using this sort of notation if it's not common in your context. – Mark S. Apr 09 '19 at 11:48