Ok so I'm learning about partial orders, and I think I know the difference between a minimum/minimal & maximum/maximal element of a partially ordered set... but I just was wondering if someone could see if my examples are correct or incorrect and why.
1) For the power set of {1, 2, ..., n}, where n is a positive integer, under the empty relation, would the minimal and minimum element be 1? Would the maximal and maximum be n? (b.t.w what does "under the empty relation" even mean?)
(In this case maximum = maximal and minimum = minimal, right?)
2) For the set of nonnegative integers under divisibility, would the minimum/minimal be 0 and there would be no maximal/maximum? (In this case maximum = maximal and minimum = minimal, right?)
3) Is it possible to have a set with no minimal element but a minimum element? Can you please give an example? This makes absolutely no sense to me.
Thanks!