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For $x_1, x_2$ the maximum and minimum are related by the equation $$ \min\{ x_1, x_2 \} = x_1 + x_2 - \max\{ x_1, x_2 \} $$ The generalization of above equation for $N > 2$ variables does not hold. However, is it possible to somehow define a relationship between minimum and maximum for $N$ variables as well?

mathcounterexamples.net
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Cesare
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4 Answers4

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The first identity follows from

$$\min\{ x_1, x_2 \} = - \max\{-x_1,-x_2 \} $$

which is equivalent to

$$\min\{ x_1, x_2 \} = x_1+x_2- \max\{x_1+x_2-x_1,x_1+x_2-x_2 \} .$$

This does not work well with three variables,

$$\min\{ x_1, x_2 , x_3 \}\\= x_1+x_2+x_3- \max\{x_1+x_2+x_3-x_1,x_1+x_2+x_3-x_2,x_1+x_2+x_3-x_3 \} $$

or $$\min\{ x_1, x_2 , x_3 \} = x_1+x_2+x_3- \max\{x_2+x_3,x_1+x_3,x_1+x_2\}. $$

A possible explanation of why you can't do better is that if you know the minimum and maximum of two variables, you know both of them and this is not true with more variables.

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You can calculate the $k$-th smallest and the $k$-th largest element of a set using $\min$ and/or $\max$.

The $k$-th largest element is

$$\max\{\min_{A\subseteq\{x_1,\ldots,x_n\},|A|=k}(A)\}$$

and $k$-th smallest is

$$\min\{\max_{A\subseteq\{x_1,\ldots,x_n\},|A|=k}(A)\}$$

These functions are related in the following way:

$$\max\{\min_{A\subseteq\{x_1,\ldots,x_n\},|A|=k}(A)\}=\min\{\max_{A\subseteq\{x_1,\ldots,x_n\},|A|=n+1-k}(A)\}$$

$$\sum_{k=1}^nx_k=\sum_{k=1}^n\max\{\min_{A\subseteq\{x_1,\ldots,x_n\},|A|=k}(A)\}=\sum_{k=1}^n\min\{\max_{A\subseteq\{x_1,\ldots,x_n\},|A|=k}(A)\}$$

so for $n=3$ we can state

$$x_1+x_2+x_3\\=\max(\{x_1,x_2,x_3\})\\+\min(\{\max(\{x_2,x_3\})+\max(\{x_1,x_3\})+\max(\{x_1,x_2\}))\\+\min(\{x_1,x_2,x_3\})$$

Similar equations can be generated for arbitrary $n$.

miracle173
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  • Thanks! Has that something to do with the Möbius inverse? – Cesare Jun 21 '20 at 14:27
  • Isn't that somehow also related to the inclusion-exclusion principle? https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle – Cesare Jun 21 '20 at 15:09
  • @Cesare: I don_t think so, but I don't know. I started with a recursive definition of the the k-th largest element and ended here. You can see this in version 5 of the edit history. – miracle173 Jun 21 '20 at 18:21
  • I think that the relationship with the inclusion-exclusion criterion is indeed there. – Cesare Jun 21 '20 at 18:58
  • Here: https://math.stackexchange.com/questions/2578996/relation-between-inclusion-exclusion-principle-and-maximum-minimums-identity – Cesare Jun 21 '20 at 18:59
  • I had no idea that this existed: https://en.wikipedia.org/wiki/Maximum-minimums_identity – Cesare Jun 21 '20 at 20:55
  • I had no idea, too, that these identities based on the inclusion-exclusion principle exists. But I cannot see that they are related to the identities here. – miracle173 Jun 21 '20 at 23:26
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The first identity is just a happy coincidence for small sets because $\min$ is just one of the numbers, and $\max$ is the other (if they are equal it's trivial too) and one element is the sum minus the other one.

I don't think there is an easy generalisation with sums will hold for larger sets. Not everything generalises.

Henno Brandsma
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The solution can be found here. I didn't know about its existence. Here some more information on its relationship to the inclusion-exclusion principle.

Cesare
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