- The relation is reflexive: for any real number , we have .
- The relation is anti-symmetric: for any two real numbers and we have:

- The relation is transitive: for any three real numbers , and ,
we have:

such that |

The relation is an ordering on :

- The relation is reflexive: for any , thus .
- The relation is anti-symmetric: for any two numbers in
, we have:

It follows that , thus . As and are positive integers, the only possibility is , i.e. . - The relation is transitive: for any three numbers in
, we have:

It follows that . As and are positive integers, the product is a positive integer and .

- (i)
- Draw the total diagram of the ordering (the lower the element is displayed as a vertex, the smaller this element is with respect to the ordering);
- (ii)
- Erase the loops on the vertices;
- (iii)
- Erase the composite arrows, i.e. the arrows which can be obtained by transitivity from more than one arrow.