Equivalence Relation Induced by Preordering/Examples

Examples of Equivalence Relations Induced by Preorderings

Finite Set Difference on Natural Numbers

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.