Partially Ordered Set, example from Wiki.
I was reading in the Naive Set Theory book by Halmos, and encountered partially ordered sets.
It seems like a lot of people understand these things immediately. I am not one of those :)
I did a bit of searching and read through the wiki on posets. Here's an example they give:
Set of natural numbers equipped with the relation of divisibility.
is the set of all positive divisors of 60.
Then this relation (divisibility) is reflexive, transitive and antisymmetric.
As I understand it, the ordered pair is in a partially ordered subset of .
Now, I guess I can "create" these ordered pairs by taking the Cartesian Product, .
This gives me among other things the diagonal terms , all in all 11 diagonal terms.
I can then go ahead and create a subset of A that is reflexive:
There are then off-diagonal elements, so there are subsets of the off-diagonal pairs.
I can combine any of these subsets with the diagonal to get a reflexive relation on A.
All in all, reflexive relations..
Is this a good way to think of the reflexive property, or am I way off?