If we have sets, , and we form a set by taking the sum of each element in the set with each element in the other sets, what can we say about the mode of ?
Intuitively, I would like to think that we can simply take the sum of the modes, i.e:
However, this seems unlikely, especially as we would expect that could potentially be a set of values, rather than a single value.
So I was wondering if we'd be able to relax this condition to state that , where we define as the set formed by taking the sum of each element in with each element in , formally:
This seems to be true, but I was wondering if we could say anything stronger?