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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.

1 Answers 1