If we have N sets, <mo fence="false" stretchy="false">{ A 1 </msub>

Banguizb

Banguizb

Answered question

2022-07-09

If we have N sets, { A 1 , , A N }, and we form a set S 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 S?
Intuitively, I would like to think that we can simply take the sum of the modes, i.e:
Mode ( S ) = n = 1 N Mode ( A n )
However, this seems unlikely, especially as we would expect that Mode ( A n ) 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 Mode ( S ) n = 1 N Mode ( A n ), where we define Mode ( A ) + Mode ( B ) as the set formed by taking the sum of each element in Mode ( A ) with each element in Mode ( B ), formally:
Mode ( S ) n = 1 N Mode ( A n ) = { n = 1 N x n : x i A i }
This seems to be true, but I was wondering if we could say anything stronger?

Answer & Explanation

Darrell Valencia

Darrell Valencia

Beginner2022-07-10Added 10 answers

Consider the multisets A = { { 0 , 2 , 2 , 3 } } and B = { { 0 , 2 , 2 , 3 , 5 } }, both with mode 2. Then
A + B = { { 0 , 2 , 2 , 2 , 2 , 3 , 3 , 4 , 4 , 4 , 4 , 5 , 5 , 5 , 5 , 5 , 6 , 7 , 7 , 8 } } ,
whose mode is 5.

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