Problem 1: Let $S$ be an ordered set with a unique maximal element $x$.
Again, I've worked some thoughts to these problems and would like to confirm their validity. I appreciate any feedback.
(1) Prove that if S is finite, then $x$ is the last element of $S$.
For $x$ to be a last element, it must strictly succeed every other element, but I am not sure how to do the proof - I thought of this in a calculus-based context of a local maximum.
(2) When S is infinite, is it true that $x$ is the last element of $S$?
I am wondering if I have to divide this into 3 cases where:
- The infinite set has a first element but no last element.
- The infinite set has a last element but no first element.
- The infinite set has neither a first nor a last element.
Problem 2: Let $S$ be a set with 5 elements.
(1) How many different linear orders are there on $S$?
I claimed that we would have 5! = 120 linear/total orders.
(2) Find the number of distinct partial orders on $S$ that have both first and last elements.
I claimed that we would have: (5 choose 2) * 3! = 60 of these partial orders.