How can I find the maximum/minimum and maximal/minimal elements of a poset?

Question

My teacher has given us really unclear definitions for all these terms, and now I have this assignment due where I have to find the maximum, minimum, and maximal/minimal elements of this poset: $$A=\{2, 3, 4,...,17\}$$ and define $\le$ on $A$ by $a\le b$ if $ a|b.$ How?? Help! (Sorry I don't know how to make the funny less than or equal to sign)

Answer 1

Learn by playing with it.

Take a piece of paper and start drawing! Put a $1$ at the bottom and just start drawing arrows. You put an arrow whenever one number evenly divides into another, for example$$1\rightarrow 2\rightarrow 4 \rightarrow 8 \rightarrow 16$$ should be in your drawing.

If you manage to draw all the possible arrows you have completely drawn out your poset. Some element will be minimal, in that it will be the smallest possible and there will be a path to every other element.

A maximal element is one that does not point to anything else. So you look at your drawing and just find those.