If min and max are only used on ordered sets, all reasonable definitions are equivalent.
In practice, however, min and max are used on preordered sets: sets in which you can have two elements that sort the same without being identical. For example, you could be manipulating:
struct student {
char *name;
int grade;
};
and define s1 < s2 when strcmp(s1->name, s2->name) < 0. Then two students with the same name but different grades will sort the same. Such two elements are said to be equivalent for the (pre)ordering relation.
On a preordered set, the argument goes, min of two equivalent elements should return the first parameter, and max should return the second. This definition preserves a few properties that you'd expect, most notably
- the pair (
min(x,y), max(x,y)) is either (x,y) or (y,x),
and
- if
x and y are distinct, then min(x,y) and max(x,y) are distinct,
and
- the function that maps (
x,y) to (min(x,y), max(x,y)) is a stable sorting function for sets of two elements.
This is not a new idea, and you'll find way better explanations than mine in a number of standard texts on programming. Chapter 7 of the Stepanov Papers, already cited by Mat and juanchopanza, is a good source if you like C++ syntax.
