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It's common practice to use ∈ to denote set membership, such as 2 ∈ A where A is a set: A=(1,2,3,4,5). This just states 2 is an element of set A.

However, is it improper to use ∈ in the context of a subset, namely to state that a contiguous subset is included in a larger set? For example a subset:

X = (1,2,3)

A = (1,2,3,4,5)

Would it be incorrect to say X ∈ A?

Jamerson2
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  • Why is $X \subset A$ not satisfactory? – Henry Dec 11 '20 at 00:11
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    Yes, it is incorrect. If you really mean that $X$ and $A$ are just sets, the correct notation is $X\subseteq A$ (or, if you want to make it clear that $X$ is a proper subset of $A$, $X\subsetneqq A$ or the like). – Brian M. Scott Dec 11 '20 at 00:13
  • Thanks, that almost exactly what I need, but I'd also need to indicate that the proper subset is contiguous and not just in relative order. How would I indicate that? – Jamerson2 Dec 11 '20 at 00:17
  • I see. Is there a standard way to denote a contiguous subsequence of a larger sequence? – Jamerson2 Dec 11 '20 at 00:22

2 Answers2

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It is incorrect, full stop. If you really mean that $X$ and $A$ are just sets, the correct notation is $X\subseteq A$ (or, if you want to make it clear that $X$ is a proper subset of $A$, $X\subsetneqq A$ or the like). If $X$ and $A$ are sequences, or an ordered triple and an ordered $5$-tuple, there is no standard notation; if you need to talk about this kind of relationship, you should define a suitable notation. Here, for instance, you might for instance say that $X=A[1,3]$. Or you might define $\preceq$ to mean subsequence of and write $X\preceq A$.

Brian M. Scott
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It's not too improper if you clearly state what you mean in your context by the $\in$ symbol. But viewed differently, yes, it is improper because $\in$ has a well established standard meaning.

So I think it's way more appropriate to assign a new symbol for your purpose, and just define its meaning rigorously and use it.

You didn't define this really rigorously since sets have no order, you probably meant sequences here.

peter.petrov
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  • Thank you. Is there a common symbol that is used for such assignments that isn't easily confounded with established meanings? – Jamerson2 Dec 11 '20 at 00:01
  • There's no standard notation - see here: https://math.stackexchange.com/questions/91384/notation-for-a-subsequence-of-a-sequence So you are free to just pick some symbol that you like. – peter.petrov Dec 11 '20 at 00:03
  • Thanks for the help! – Jamerson2 Dec 11 '20 at 00:03
  • Actually a person there suggests the notation $S'|S$, seems that symbol is already used (in certain texts) to mean just that: $S'$ is subsequence of $S$. Here is that answer: https://math.stackexchange.com/a/2573874/116591 – peter.petrov Dec 11 '20 at 00:04