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The maximum-minimums identity states that

$$\max(a,b,c) = a+b+c - \min(a,b) - \min(a,c) - \min(b,c) + \min(a,b,c).$$

Now I "corrupt" some of the right hand side like so

$$LHS = a+b+c - \min(a,b,d) - \min(a,c,e) - \min(b,c,f) + \min(a,b,c,d,e,f).$$

What is a new expression for the LHS?

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Just wasn't sure what other tag I could use, seemed to be the closest match I could think of. Any suggestions? – Projectile Fish Jul 25 '11 at 10:15
@Projectile Fish: [algebra] tag or one of its variations (perhaps [algebra-precalculus]?). While it is true that the vast majority of modern mathematics can be developed within ZFC, and perhaps a good question fitting [set-theory] would be how to do so, this is certainly not a set theory related question. In simpler terms: not all questions with sets in them involve set theory. – Asaf Karagila Jul 25 '11 at 10:29
up vote 2 down vote accepted

I don’t see any reason to think that there is a nice expression for the new RHS. Suppose that $a$, $b$, and $c$ are the smallest of the six numbers; then the RHS is $$a+b+c-\min\{a,b\}-\min\{a,c\}-\min\{b,c\}+\min\{a,b,c\}=\max\{a,b,c\},$$ as before. If, on the other hand, $a$, $b$, and $c$ are the largest of the six numbers, then the RHS is $$a+b+c-d-e-f+\min\{d,e,f\},$$ so it’s one of the numbers $a+b+c-d-e$, $a+b+c-d-f$, and $a+b+c-e-f$, none of which need have much of anything to do with $\max\{a,b,c\}$. Other relationships between $a$, $b$, and $c$ on the one hand and $d$, $e$, and $f$ on the other lead to even more variety.

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