Check union closedness of set

Question

I have defined two sets

set1[n_] := Table[i, {i, 1, n}];
set2[n_] := Subsets[Delete[Subsets[set[n]], 1]];

Now, I want to check whether an element of set2[n] is union closed, i.e. if the union of every two elements in set2[n] is again an element of set2[n].. How can I realize this?

Answer 1

An ugly solution, test them one by one:

closeness[set_]:=Block[{check=True},Do[If[FreeQ[set,Union[set[[i]],set[[j]]]],check=False;Break[]],{i,1,Length@set},{j,i+1,Length@set}];check]