Partial Order relation conditions

Question

A relation is Partial Order relation if it is

1. Reflexive

2. Anti-Symmetric

3. Transitive

I have never seen (example) Anti-Symmetric rule playing a decisive role in Partial Order relation as it is some what similar to Reflexive.

Can you provide an example clearing this doubt?

Answer 1

If a relation is symmetric, then you have $xRy$ if and only if $yRx$ ... that's not much of an 'order', is it? That is, you couldn't say that $x$ is ordered 'before' some other element $y$. Indeed, a relation that is reflexive, transitive, and symmetrical would be an equivalence relation, which is pretty much the opposite of what you'd want for an order.

So, by making the relation anti-symmetric (or asymmetric, to get a strict order), you rule out such possible symmetry, and you can actually start talking about an order.