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:

\(\ds \)

\(\)

\(\ds a \sim_\RR b\)

\(\ds \)

\(\leadstoandfrom\)

\(\ds a \mathrel \RR b \land b \mathrel \RR a\)

Definition of Equivalence Relation Induced by Preordering

\(\ds \)

\(\leadstoandfrom\)

\(\ds a \setminus b \text { is finite } \land b \setminus a \text { is finite }\)

Definition of $\RR$

\(\ds \)

\(\leadstoandfrom\)

\(\ds \paren {a \setminus b} \cup \paren {b \setminus a} \text { is finite }\)

Definition of Finite Set

\(\ds \)

\(\leadstoandfrom\)

\(\ds a \symdif b \text { is finite }\)

Definition of Symmetric Difference

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

$\blacksquare$

Sources

• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations: Exercise $39 \ \text{(b)}$