The question I have problem with is
a) How many partial orders for the set {x,y} has x as a minimal element
b) How many partial orders for the set {x,y,z} has x as a minimal element
I don't really get how I should think. I believe I understand how it works when you use a diagram, but now I can't get a grip on how the diagram would look for a problem like this. Or can't you use that way of thinking in this problem?
a. There are three orders for {x,y}, x < y, y < x and
{x,y} is an antichain (x and y cannot be compared, x||y).
Which of those orders have x as a minimal order?
b. What are the possible orders for {x,y,z}?
If you must have diagrams, then draw a diagram for each order.