Equivalence Relation Induced by Preordering/Examples/Finite Set Difference on Natural Numbers

Example of Equivalence Relation Induced by Preordering
Consider the preordering $\RR$ 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.

Let $\sim_\RR$ denote the equivalence relation on $\powerset \N$ induced by $\RR$.

Then the $\sim_\RR$-equivalence class of an element $a$ of $\powerset \N$ is defined as:
 * $\eqclass a {\sim_\RR} = \set {b \in \powerset \N: a \symdif b \text { is finite} }$

where $\symdif$ denotes symmetric difference.

Proof
We have that:

The result follows by definition of $\sim_\RR$-equivalence class.