Preordering/Examples

Examples of Preorderings

Finite Set Difference on Natural Numbers

Consider the relation $\RR$ on the powerset of the natural numbers:

$\forall a, b \in \powerset \N: a \mathrel \RR b \iff a \setminus b \text { is finite}$

where $\setminus$ denotes set difference.

Then $\RR$ is a preordering on $\powerset \N$, but not an ordering on $\powerset \N$.