Hi,

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.

$\displaystyle A=\{1,2,3,4,5,10,12,15,20,30,60\}$ is the set of all positive divisors of 60.

Then this relation (divisibility) is reflexive, transitive and antisymmetric.

(1)Reflexive: $\displaystyle aRa,\;for\;all\;a\in A$

So, $\displaystyle (a,a)\in R$

As I understand it, the ordered pair $\displaystyle (a,a)$ is in a partially ordered subset of $\displaystyle A$.

Now, I guess I can "create" these ordered pairs by taking the Cartesian Product, $\displaystyle AxA$.

This gives me among other things the diagonal terms $\displaystyle (1,1),(2,2),(3,3),\cdots,(60,60)$, all in all 11 diagonal terms.

I can then go ahead and create a subset of A that is reflexive:

perhaps, $\displaystyle \{(1,2),(1,1),(2,2)\}$

or, $\displaystyle \{(5,3),(5,5),(3,3)\} $

There are then $\displaystyle n^2-n$ off-diagonal elements, so there are $\displaystyle 2^{n^2-n}$ 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, $\displaystyle 2^{n^2-n}$ reflexive relations..

Is this a good way to think of the reflexive property, or am I way off?

Thanks!