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Let $U$ be a certain function on sets ($U(S)$ represents the 'value' of the set $S$).

If a set $S$ is partitioned to two disjoint subsets $A$ and $B$ with $A \cup B=S$, then: $V(A,B):=U(A)+U(B)$ ($V$ represents the 'value' of the partition $(A,B)$).

Now I would like to define a function $W$ that represents the "value of the best partition":

$$W(S) := \max_{(A,B)\ is\ a\ partition\ of\ S}V(A,B)$$

My question is: does it make more sense to use $\max$ or $\sup$ in the definition? I have read about the Difference between supremum and maximum but I am still not sure what will be the consequences of using "max" vs. "sup" in the above definition.

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Erel Segal-Halevi
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1 Answers1

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I think it depends on the size of your set S.

If it is finite, feel free to use maximum (as you can then just "list all the partitions, find the most valuable one and write down its value", i.e. the value is attained).

If it is not finite, I would go for supremum - the least upper bound $M$ does not have to be attained for any partition (A,B) (but there can be a sequence of partitions $(A_n, B_n)$ s.t. $V(A_n,B_m) \rightarrow M$).

Sorry if I'm not correct!

njaja
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