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In one of my other questions (which has no answers by the way - I admit it's rather difficult!), I define a matrix in which each entry is a set. Now that I think about it, I wonder if defining a matrix with set entries is bad practice, because it's not like I can use typical linear algebra tools. I haven't seen many people define matrices like that in past. On the other hand, defining it as such helps me understand the problem a little better. The nice thing about a matrix is it has order in two directions, and it's easy to reference a specific entry, which is helpful in the context of my problem.

Should I avoid doing this practice? Would the Math Illuminati frown upon it?

Here's the link to the question I mention: Equilibrium existence proof

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Shane
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    The question is not whether or not it's bad practice, but what is it good for? – Asaf Karagila Oct 27 '14 at 14:23
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    Also there's no such thing as the Math Illuminati. It's the Math Freemasons. Why do you think there are so many "free" objects in mathematics, and so many things which are "constructed"? – Asaf Karagila Oct 27 '14 at 14:27
  • Well - it's a way to put information into a nice, easily referenceable rectangle. That's what it's good for. I guess I don't see why the idea of putting information into rectangles should be reserved for those seeking to then do operations on those rectangles! Sorry, Math Freemasons, yes - that makes much more sense. – Shane Oct 27 '14 at 14:27
  • If the Math Police comes to take you away, you could always save your neck by claiming it is a matrix over the ring of subsets of $U$, with symmetric difference and intersection. – hmakholm left over Monica Oct 27 '14 at 14:30
  • @Henning - I have no idea what that means. But then again, neither will my advisor, so that might well work. – Shane Oct 27 '14 at 14:33

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You can certainly define a family of sets indexed by $\{1,\ldots,n\}\times\{1,\ldots,m\}$, and you can write them in a rectangular structure if you have a need to. Nothing at all wrong with this. (And you might well use LaTeX's matrix environments to typeset it).

But it probably wouldn't do much mathematical good to call this family a "matrix", unless you have some use for the idea of matrix multiplication, which is the defining feature that makes a matrix a matrix rather than just a two-dimensional array of things.

hmakholm left over Monica
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    In some aspects you could argue that every matrix is a matrix of sets, since everything is a set! :-) – Asaf Karagila Oct 27 '14 at 14:27
  • OK - I think that answers my question. I should just be calling it a family of indexed sets. This terminology was unfamiliar to me. Thanks! – Shane Oct 27 '14 at 14:32