Designing less than and less than or equal operators for Map with value being a Set
I think that actual code is off-topic here.
In general you can define a strict total order relation on (sub)sets of a known universe $U$: pick an arbitrary order for the elements of $U$ then, given two distinct subsets $X,Y$ of $U$, you can compare $X$ and $Y$ by first sorting the elements of $X$ (resp. $Y$) and then comparison the sorted version of $X$ with the sorted version of $Y$ lexicographically.
In your particular example the sets seem to contain integers, so you can just sort the set using the standard &quot;less than&quot; relation between integers.
To extend this total order to a map you can notice that a map is essentially a container of pairs of the form (key, value). We just defined a linear order over the values (sets), and you probably already have a natural order over the keys. Then two (key, value) pairs can be again compared lexicographically.