I just started Analysis and I have a few questions,
1. "In a partial order on , an element is maximal if , and is maximum if for all "
It seems to me that these definitions are equivalent, since if it means that such that , so x is bigger than every other element in the set.
Also, is pretty much saying x is bigger than every other element in the set, right?
So what am I missing? :/
"A subset Y of X such that for any , either or is called a chain. If X itself is a chain, the partial order is a linear or total order."
Just to make sure I'm understanding this alright:
Say you have a set
Then would with partial order and with partial order be "chains"?
What are other partial orders you could have? I can only think of
3. Do partially ordered sets have to be countable?
4. What does mean?