Min and max in order of operations

In summary, the conversation discusses the order in which to apply min and max in the expression min_{i \in I}max_{j \in J} (set of elements with indices i \in I and j \in J). The usual convention is to evaluate from right to left, but it is important to clarify with the author for confirmation. The expression can also be written as min max (set of elements) and the author's website allows for a free download of the paper. The order of operations is important as shown in the example of taking the min of max versus the max of min. There is a theorem that states that the min of max is greater than or equal to the max of min.
  • #1
eyec
6
0
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/teaching/cs532a%20-%202003-4/folk.pdf [Broken] p.1 which refers to in http://dx.doi.org/10.1016%2F0022-0531(79)90002-4 [Broken] (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 ...
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

1. What is the purpose of using min and max in order of operations?

The purpose of using min and max in order of operations is to determine the smallest (minimum) or largest (maximum) value among a set of numbers or variables. This is important in many mathematical and scientific calculations, as well as in programming and data analysis.

2. How do you use min and max in an equation?

To use min and max in an equation, you would first identify the set of numbers or variables that you want to compare. Then, you would simply replace the min or max function with the appropriate value in your equation. For example, if you have the equation y = min(x, 10), you would use the smaller value between x and 10 as the value for y.

3. Can min and max be used with more than two values?

Yes, min and max can be used with any number of values. For example, you can have an equation like z = max(a, b, c, d) where z will be assigned the largest value among a, b, c, and d. This is useful when comparing multiple sets of data or when finding the extremes in a large dataset.

4. What is the difference between min and max?

The main difference between min and max is that min returns the smallest value among a set of numbers or variables, while max returns the largest value. In other words, min is used to find the minimum value and max is used to find the maximum value.

5. Are min and max affected by the order of the numbers or variables?

No, min and max are not affected by the order of the numbers or variables. They will always return the same value regardless of the order in which the numbers or variables are listed. For example, min(5, 10) and min(10, 5) will both return 5 as the minimum value.

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