For example:
Let $\mathrm{S}$ be a 2x2 square matrix. Each element in $\mathrm{S}$, denoted by the term $\mathrm{sector_{\alpha \beta}}$, is itself a set of non-numerical objects (a custom software type). Let us use letters to represent these objects (whose contents/value is fixed).
$$\mathrm{S} = \begin{bmatrix} \mathrm{x} & \mathrm{a,b,c} \\ \mathrm{y} & \mathrm{z} \\ \end{bmatrix}$$
So, $\ \mathrm{sector_{11}} = \mathrm{\{x\}}$, $\ \mathrm{sector_{12}} = \mathrm{\{a,b,c\}}$, $\ \mathrm{sector_{21}} = \mathrm{\{y\}}$, $\ \mathrm{sector_{22}} = \mathrm{\{z\}}$
(Okay, I know I could have just used $\ \mathrm{S_{\alpha \beta}}\ $ instead of $\ \mathrm{sector_{\alpha \beta}}\ $ but indulge me...)
Is a $\ \mathrm{sector}\ $ a subset of $\ \mathrm{S}\ $ or an element of $\ \mathrm{S}$ ? I suppose it depends on whether you can consider a matrix a special sort of set, or an object type only? I mean, $\mathrm{S}$ is itself a subset of $\mathbb{M_n}$
This is just a fun question for my own curiosity, because I actually get off on the terminology of mathematics. It's a sickness I am sure at least some of you share with me! (It occurred as I am in the middle of formulating a description of a solution using arrays within arrays whose elements are slices of other, larger arrays! Fun!)
P.S. What other contexts might such an enjoyable ambiguity of membership exist? If there even is any ambiguity at all, I could just be completely misconceiving this whole thing...
In set theory, a set $x$ is transitive if for all $y$, if $y\in x$ then $y\subseteq x$. The (von Neumann) ordinals are defined to be the transitive sets that are well-ordered by $\in$. Thus the "$<$" relation between ordinals is simply $\in$. If $\alpha, \beta$ are ordinals with $\alpha\in \beta$, then $\alpha \subseteq \beta$. So any ordinal is a fine example of a superset, some of whose subsets are also members (namely, every smaller ordinal is both a member and a subset).