# Succinct notation for finite sets

I have sometimes somewhere see a very concise way to express sets of finite size whose elements are really not important, only that they are different, but I'm not sure what of the following variants it was (or maybe there is even better notations):

$$S=\{s_i\}\\ S=\{s\}_i\\ S=\{s_i\}_1^n$$

I hope it's the first notation $$S=\{s_i\}$$, because you give the same information as in the other methods but write much less; you express the index you will use whenever you refer to elements in $$S$$, you don't need to iterate from $$1$$ to $$n$$ because in maths, indexes usually start at $$1$$, and the last index, or number of elements, can always be written as $$\#S$$ or $$|S|$$, so taking the symbol $$n$$ is unnecesary.

As a related question, when you want to express set of pairs, what you would write, assuming one of the above notations are correct?

$$S=\{(a, b)_i\}, \text{ or}\\ S=\{\{a, b\}_i\}, \text{ or even}\\ S=\{a, b\}_i$$

• With the first notation, nothing prevets from having $i \in \mathbb N$ Dec 31, 2020 at 11:13
• I consider the first two absolutely unacceptable: the first denotes a singleton, and the second gives no indication of the set over which the index ranges, which might well be infinite. The last is used, but I consider it substandard and would never use it. If all that matters is that $S$ is a subset of $X$ of cardinality $n$, write $S\in[X]^n$. If all that matters is that the cardinality of $S$ is finite, you can write $S\in[X]^{<\omega}$, or $S\in\wp_{\text{fin}}(X)$, or the like. Dec 31, 2020 at 20:07

Without context, I would interpret your first three definitions of $$S$$ as:

1. A set consisting of a single element called $$s_i$$,
2. A set of elements $$s$$, all identical, indexed by $$i$$ which is probably, but not definitely, infinite, and
3. A finite set of $$n$$ elements $$s_1, s_2, \ldots, s_n$$ which may be all different or may contain repeats.

In cases $$1$$ and $$2$$ I would look for further context nearby where the symbols are introduced and worry if I couldn't find it; for $$3$$ I would feel I already understand from what is written.

The succinct form of the set is $$S$$, in fact, so it is important to define what you mean when $$S$$ is introduced, and then you can refer back to just $$S$$ whenever you need it. To specify what you're after: a finite set of distinct elements, I would suggest: $$S := \{ s_i \mid s_i \not= s_j, 1\leq i,j \leq N \}$$ where we assume that the reader is not about to assume that $$s_1 \not= s_1$$ just to be difficult.

If you need to refer to several sets of this kind, I would set that up as a notational paragraph at the start; something like:

We define $$S_N, T_N$$, etc. to be any set of $$N$$ distinct elements from an alphabet/universe $$\cal U$$, specifically $$S_N := \{ s_i \mid s_i \not= s_j, 1\leq i,j \leq N \}$$

If using the notation you've described is really important to you, then again, define what you mean by it and then probably use $$\{s_i\}_{i \leq N}$$ as the short form.

• Yes I think I will go for you last choice, specify at the start what do I mean with the notation I want to use and then just use it once clarified.
– ABu
Dec 31, 2020 at 11:16
• I think that is what I was looking for en.wikipedia.org/wiki/Indexed_family#Mathematical_statement
– ABu
Jan 1, 2021 at 12:40