# Partially ordered set proof

I'm trying to proof if the following Relations R ⊆ M×M total order or partially order are.

1. $M = \{1,2,3\} , R = \{(x,y) : x|y\}$

2. $M = {\bf Z} , R = \{(x,y) : x\vert y\}$

3. $M = {\bf N}, R = \{(x,y): y ≤ x\}$

4. $M = {\bf Z} × {\bf Z}, R = \{((x1,x2),(y1,y2)):x1 ≤ y1 ∧ x2 ≤ y2\}$.

Well i know exactly how to determine if the relation total order or partially order

Total order is reflexiv, transitiv, antisymmetric and total/linear.

Partially total order is reflexiv, transitiv and antisymetric.

I have some ideas for these relations but to be honest i have no idea about how to proof it.

Well if a relation would given for example like this :

M = {1,2,3} , R = {(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)}

I could say that this is a total order, i see exactly the all binary couples in the relation. But the exercises above confuse me , because i don't see the set(or binary couples). There are just some conditons. For exampe in 1) it says a divides b. But how can i proof using this condition if this relation is reflexiv, antisymmetric and so on. This is my problem !!!

3x3 table for the first one looks like this: $\{(x,y) \in M\times M : x | y\} = \{(1,1), (1,2), (1,3), (2,2), (3,3)\}$, that is, $$\left[\begin{array}{ccc} 1&1&1\\0&1&0\\0&0&1\end{array}\right].$$ Try doing it yourself for the rest of the exercise. – dtldarek Dec 16 '12 at 17:48