min and max in order of operations

eyec

hi,

i was unfortunately unable to find any information about the order of operations in which to apply min and max. (or is it just common knowledge for everyone except me?) consider

[itex]min_{i \in I}max_{j \in J}[/itex] (set of elements with indices [itex]i \in I[/itex] and [itex]j \in J[/itex])

(actually, this is from http://www.cs.ubc.ca/~kevinlb/teachi...003-4/folk.pdf p.1 which refers to in http://dx.doi.org/10.1016%2F0022-0531(79)90002-4 (which is, however, not free), p.4.) is the minimum or the maximum function applied first? the context in the mentioned papers seems to imply that the first one (min) has precedence. This, however, does not seem intuitive to me because then we have a notation "operation1 operation2 arguments" and do operation2(operation1(arguments)).

thanks ...

chiro

Hey eyec and welcome to the forums.

The usual convention for this kind of thing is to evaluate things from right to left, but if in doubt you should consult the author for clarification.

So with the usual convention, max is evaluated first and then min.

eyec

thanks chiro,
so I'm at least certain that I'm not completely wrong about how to deal with max, min ... etc.

maybe i'll ask the author ... in case I find out something, I'll post that again.

AlephZero

min and max must act on a collection of objects. So there is only one way to iterpret your expression that makes any sense, and that is

min ( max ( set of elements ) )

You normally write it as

min max ( set of elements )

because the outer ( )s don't add any more information about what it means.

You usally need ()'s around the "set of elements", because the notation for the set is often fairly long and complicated and it's not obvious where it ends.

This is no different from writing say
log sin (x/2)
You wouldn't normally write
log (sin (x/2))
because that's the only thing the expression could mean. But of course
log sin (x/2)
and
sin log (x/2)
mean two different things.

eyec

thanks AlephZero,

you are certainly right (& so is chiro).

probably the author also meant it like that. if someone wants to examine it, on sciencedirect the paper costs a lot of money but on the author's website you can download it for free. (I just realized that now.) http://arielrubinstein.tau.ac.il/papers/03.pdf
the equation in question is on page 4, first line of section 3.

haruspex

The order is important. Consider
1 2 3
4 2 0
3 2 1
If I take min in each col I get 1, 2, 0, the max of which is 2.
If I take the max of of each row, I get 3, 4, 3, the min of which is 3.
There's some theorem that says min of max >= max of min.
As others have noted, right-to-left is the only valid interpretation.