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My question is related to set notation.

How can I differentiate with proper mathematical notation between the following 2 subset selection approaches?

From an initial set $S=\left\{q_{i}\right\}_{i=1}^{K}$ with $K$ items, with $q_{i}\in\mathbb{R}$, I want to select one subset $s\subset S$ with $N$ items ($N<K$) out of the total $K$. The selection could be

  • A. Random. For example if $S=\left\{10,2,35,14,58\right\}$ with ($K=5$) and I want to select a subset with $N=3$ items, some possible cases could be $s=\left\{10,14,58\right\}$, $s=\left\{2,35,58\right\}$, $s=\left\{10,2,14\right\}$ etc.

  • B. Sequential. For example, if $S=\left\{10,2,35,14,58\right\}$ with ($K=5$) and I want to select a subset with $N=2$ items, the result would be $s=\left\{10,2\right\}$, for $N=3$ it would be $s=\left\{10,2,35\right\}$, and for $N=4$, $s=\left\{10,2,35,14\right\}$ and so on.

How can I express these 2 cases with proper math notation?

What comes to my mind is simply $s=\left\{q_{i}\right\}_{i=1}^{N}$, which seems to me more like case B. Is that correct? If yes, then how can I express case A and how will this be different from case B?

P.S. There is no specific formula to describe the elements of the initial set $S$. It could be any numbers in $\mathbb{R}$.

robjohn
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user53785
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1 Answers1

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I'm not sure I've seen explicit notation for this, but what I would usually see is something like:

Let $S_m$ be a subset of $S$ with cardinality $m$, containing any $m$ elements $s_i \in S$.

I don't think you have to specify a random ordering; this allows for any ordering. On the other hand, if you want to explicitly specify sequentiality, I'd say something like:

Let $S_m$ be a subset of $S$ with cardinality $m$ such that, if $q_i \in S_m = s_j \in S$, $q_{i+1} \in S_m = s_{m+1} \in S$. (That is, the subset $S_m$ preserves the ordering of elements in $S$.)

Edit: For the second case, you could even say something like...

Let $S_m$ be a subset of $S$ with cardinality $m$ such that $S_m = \{q_i | q_i \in S_m = s_j \in S \implies q_{i+1} \in S_m = s_{j+1} \in S\}$

AmagicalFishy
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  • Thnx very much for the answer. I think that the main point here is that it skipped my mind that in set theory order is irrelevant. Thus what you wrote for case A it is correct and I prefer it from what I had in mind. Regarding case B, I think what I am describing is closer to a n-tuple, where ordering matters. Thnx again for the help. – user53785 Mar 04 '15 at 02:56