How to do the 3 checks on a partially ordered set

Question

I have a task to check if a relation is a partially ordered set. I know i have to check if it's:

let $S$ be a set and $\mathcal{R} $ is a relation:

1. Reflexive ( $ (s,s) \in \mathcal{R} \forall s \in S$ )
2. Antisymetric ( $\forall s,t \in S: (s,t) \in \mathcal{R} \space and \space (t,s) \in \mathcal{R} \space \Rightarrow s = t $ )
3. Transitive ( $ \forall r,s,t \in S: (s,t),(t,r) \in \mathcal{R} \space \Rightarrow (s,r) \in \mathcal{R} $ )

But i'm having some troubles actually doing this. I am confusing myself. Can you help me going by maybe checking the reflexive property?

I have to check:

$ \mathcal{R} = \{ (m,n),(m',n') \in \mathbb{Z} \times \mathbb{Z} : n < n' \space or \space (n=n' \space and \space m \leq m') \}$

I would very much appreciate your help. Thank you.

Answer 1

Your relation is reflexive because, if $(m,n)\in\mathbb{Z}^2$, then $n=n$, and $m\leqslant m$. Therefore, by definition of $\mathcal R$, $\bigl((m,n),(m,n)\bigr)\in\mathcal R$.

Can you take it from here?