Cleaner notation for cardinality of set instead of |{⋅}|

Question

I feel it's a little ugly to use the normal "absolute value" notation for the size of an anonymous set-builder set:

$$ N = |\{ x \in \mathcal{X} : f(x) \geq 0 \}| $$

Is there a preferred replacement? I feel like I've seen

$$ N = \# \{ x \in \mathcal{X} : f(x) \geq 0 \} $$

in some informal notes, but I'm not sure if it's used in formal publications.

Answer 1

First off, the "size" of set is not well-defined. "Size" could refer to any of a plethora of different things, such as cardinality, measure, or diameter. If you mean "cardinality" (which seems to be the intended meaning, based on the the use of $\#\{x\in \mathcal{X} : f(x) \ge 0\}$), then you should say "cardinality", and not "size".

Assuming that cardinality is the meaning of the notation, then there are several notations which I have seen in the wild (in publications, on the interwebs, etc.):

• $\operatorname{card}(A)$ (this is the notation I prefer)
• $|A|$
• $\#A$
• $\mathcal{N}(A)$
• $n(A)$
• $\bar{\bar{A}}$

The notations higher on the list are, I think, somewhat more universal, and more likely to be understood from context. I would still recommend taking the space to explicitly explain the notation (e.g. "The cardinality of a set $A$ is denoted by $\#A$").

Regarding the use of these notations in formal publication, that is between you and your editor (and/or reviewers). Pick whichever notation you prefer, and change it if you are asked to by an editor or reviewer.