Find a non-convex instance for a non-convex set function with FindInstance

Question

Let $f$ be a non-convex set function defined on a set $L$. Hence there are subsets $A,B\subseteq L$ such that $$f(A)+f(B)>f(A\cup B)+f(A\cap B).$$

I tried to use FindInstance function to find a non-convex instance of $x$ and $y$ as follows:

Powerset=Subsets[L];
FindInstance[f[x]+f[y]>f[Union[x,y]]+f[Intersection[x,y]],Powerset]

But I have the following errors:

Union::normal: Nonatomic expression expected at position 1 in x\[Union]y.
Intersection::normal: Nonatomic expression expected at position 1 in x\[Intersection]y.
FindInstance::ivar: {} is not a valid variable.

Does FindInstance work with set functions? How should I modify my code?

Answer 1

findit[L_, f_] :=
 (* select pairs of subsets in the powerset Subsets[L] that satisfy the criterion *)
 ResourceFunction["SelectSubsets"][
  Subsets[L], {2}, 
  Total[f /@ #] > f[Union @@ #] + f[Intersection @@ #] &
  ]
findit[{1, 2, 3}, Median]
(* result: {{{1,2},{1,3}}} *)

(* check it works *)
Median[{1, 2}] + Median[{1, 3}] > 
 Median[Union[{1, 2}, {1, 3}]] + Median[Intersection[{1, 2}, {1, 3}]]
(* result: True *)