What is the word for set containing other sets?

Question

For example, define $S = \{\{3\}, \{2,1\}, \{1,1,1\}\}$. What is the word for $S$?

Multiset? Doesn't seem to be correct.

Edit: Also, how can I denote the cardinality of a specific set within a set?

Answer 1

Your $S$ is a set. Sets don't really much care what they contain - they're still sets. Really, a set is just an object $S$ which we interrogate about its members - we have this "membership relation" of $x\in S$ which is true if $x$ "contains" $S$ and false otherwise - and $x$ could be anything, including other sets.

It looks like you have the set of partitions of three here, if we're being very specific. Really, you ought to represent that as a set of multisets (since $\{1,1,1\}=\{1\}$ as sets). The distinct here is that a multiset $M$ not only tells you if $x\in M$, but how many of $x$ are in $M$ - so it gives a bit more information than a set (but again, it doesn't matter what it contains, so much as how it contains it).

(As to your edit, you would, for a $s\in S$, describe its cardinality as $|s|$ - it's no different from how one usually describes cardinality - you just make sure to apply it to the element rather than the whole set)

Answer 2

In ZF, every element of a set is itself a set, but we typically do not think about sets this way, e.g. $\mathbb R$. However, if a set is presented like your example, we typically call it a "set of sets", a "collection of sets", or simply a "set".

Answer 3

As written the question is confusing, since $\{1,1,1\}=\{1\}$ according to the usual conventions of set theory. One may assume that you meant the inner sets to be multisets instead, in which this equality is not true, and $\{1,1,1\}$ would be considered to have three (identical) elements, or more properly one element (namely $1$) with multiplicity$~3$. In that case your $S$ is a set of multisets.

Unfortunately there is no standard convention for multisets written by explicit enumerations of their elements. Personally I have written $\{\!\!\{1,1,1\}\!\!\}$ with double braces to signal the different interpretation (but with close spacing to distinguish from $\{\{1,1,1\}\}$ which is a singleton set containing a (in this case singleton) set, more naturally written as $\{\{1\}\}$. So you would have $$S=\{\{\!\!\{3\}\!\!\},\{\!\!\{2,1\}\!\!\},\{\!\!\{1,1,1\}\!\!\}\}$$ For the size (sum of element multiplicities) of a multiset$~M$ you could write $|M|$ or $\#M$ according to what is your conventions for ordinary sets; once it is made clear that $M$ is a multiset there should be no confusion. In the example the elements $s\in S$ have $|s|=1,2,3$, respectively.

Another notation you might sometimes need is for the multiplicity of a given element$~x$ as element of a multiset$~M$. I would suggest something like $\mu_x(M)$ so that $\mu_1(\{\!\!\{1,1,1\}\!\!\})=3$. Similarly you might sometimes need a notation for the underlying set of $M$ (an ordinary set obtained by forgetting nonzero multiplicities), for which one might suggest something like $\overline M$, but I really don't think there is much agreement about what is a good notation for this.