What's the term for this set theory operation?

Factual Questions
1

I have a question about math terminology. I can explain what I’ve done with an example, but would like someone to tell me the term for what it is I’ve done.

Say I have a set of sets S = {S[sub]1[/sub], S[sub]2[/sub], …, S[sub]m[/sub]}:

S[sub]1[/sub] = {a, b, c}
S[sub]2[/sub] = {d, e}
S[sub]3[/sub] = {f, g}
S[sub]4[/sub] = {h, d}
S[sub]5[/sub] = {i, f}

I then create a new set of sets T = {T[sub]1[/sub], T[sub]2[/sub], …, T[sub]n[/sub]} (m ≥ n) whose members unify all sets in S which share at least one common member:

T[sub]1[/sub] = {a, b, c, d, e, h}
T[sub]2[/sub] = {f, g, i}

So in set theory terminology, what’s the term for T, or for the process of finding it? I think that, by analogy with graph theory, the process might be called “finding the symmetric closure of S”, but I’m not sure if that’s correct.

2

Using your own definition, what you really mean is this for the T sets:

T1 = {a, b, c}
T2 = {f, g, i}
T3 = {d, e, h}

I don’t know why you combined the two T sets.

I don’t know of any standard term for this.

3

Sorry, you’re correct… there was a typo in the example. The line for S[sub]2** should have read as follows:

S[sub]2[/sub] = {c, d, e}

4

You’ve created unions of the intersecting subsets.
(It’s been a long time since I did this. Is Union the right word when the duplicates are eliminated?)

5

In set theory, duplicates are always eliminated. You must be thinking of ‘bags’, which are more of a computer science thing than a foundations-of-mathematics thing.

6

Thanks. Yes, I’m getting confused between set theory (long long long ago) and database set theory (not so long ago).

I’m thinking he’s making a set of subsets of unconnected unions of subsets, if that’s valid terminology. Non-overlapping? Distinct? Whatever the term is for inersecting sets, sets with members in common, versus sets with no members in common.

7

I am quite certain there is no name for such an operation. Why should there be; it doesn’t correspond to anything useful that I am aware of. If you find something useful, why then you get to name it.

It seems to me that what you are describing should be correctly be described by saying that T_n is the union of S_n and all the S_m for which m > n and S_n has non-empty intersection with S_n. Except for the m > n clause, it might be described as the star of S_n. I suppose you could call it the star of S_n in the set {S_n,S_{n+1},…}. I suppose if I had to name it, I would call it the rising star.

8

It corresponds exactly with finding the connected components in an undirected graph, which is a common operation in graph theory. It’s the same problem; just framed in a different way. That’s why I figured there would have been a name for it in set theory too.

9

Set theory geeks will love today’s XKCD.

10

That would just uee the iterated star construction. It was the m > n that confused me.

11

I’ll take a different tack in answering this question.

Bolding mine. Note that, in set theory terminology, strictly speaking, T cannot be any closure of S, because T is not a subset of the power set of S. T2 could be {S4, S5}, but not {f,g,i}. Also note that one gets a (symmetric, transitive, or whatever) closure given some relation. My guess is where you see “finding the symmetric closure of S” they mean something like “finding the symmetric closure of D, given the relation R, found by set S”, where D is the domain of objects in the graph ({a,b,c…}). It isn’t obvious to me what to plug in for R, given that this is a symmetric closure.

12

I think that if you define a relation ~ on your universe by i ~ j if and only if there’s some S which contains {i, j}, then your T sets will be the members of the quotient set with respect to the transitive closure of ~.

13

I would say that the T sets are the components of a hypergraph, where the S sets are the edges of the hypergraph:

14

Thanks! This is exactly what I was looking for.

Here’s a follow-up question: Say I have a set of sets S. Then let T = ∪[sub]X ∈ S[/sub] X, which is the set of all members of the members of S. Is there a more convenient way of writing this, and is there a name for T?