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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.

japreiss
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    I've used $#{ \ }$ notation in a paper that was published in Applied and Computational Harmonic Analysis. So it definitely can be used in a formal publication. – JimmyK4542 Dec 06 '19 at 03:59
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    Since the number of elements in a set is called the cardinality of the set, maybe writing $\operatorname{card} A$ is a good idea? – Maximal Ideal Dec 06 '19 at 04:03
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    I agree that using the "absolute value" bars is a bad notation. I use $#$ myself. – Greg Martin Dec 06 '19 at 04:19
  • I had never seen the $#{}$ notation until I moved to the United States. It puzzled me at first, then I really hated it and it drove me nuts, ... and by now I use it myself. So it may be a North American or an English-based thing. Although it's probably been picking up in popularity around the world now. Still, it feels to me that "$#{}$" is mostly used as the number of items when we can count them, i.e. for finite sets. I don't think I've ever seen it to represent the cardinality of an arbitrary set. – zipirovich Dec 06 '19 at 06:07

1 Answers1

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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.

Xander Henderson
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