Relation Induced by Partition is Equivalence

Theorem

Let $\mathbb S$ be a partition of a set $S$.

Let $\RR$ be the relation induced by $\mathbb S$.

Then:

$(1): \quad \RR$ is unique

$(2): \quad \RR$ is an equivalence relation on $S$.

Hence $\mathbb S$ is the quotient set of $S$ by $\RR$, that is:

$\mathbb S = S / \RR$

Proof

From the definition of Relation Induced by Partition, we define the relation $\RR$ on $S$ by:

$\RR = \set {\tuple {x, y}: \paren {\exists T \in \mathbb S: x \in T \land y \in T} }$

Test for Equivalence

We are to show that $\RR$ is an equivalence relation.

Checking in turn each of the criteria for equivalence:

Reflexive

$\RR$ is reflexive:

$\ds $

$\ds \paren {x \in T \land x \in T}$

Rule of Idempotence

$\ds $

$\leadstoandfrom$

$\ds \tuple {x, x} \in \RR$

Definition of $\RR$

$\Box$

Symmetric

$\RR$ is symmetric:

$\ds $

$\ds \tuple {x, y} \in \RR$

$\ds $

$\leadsto$

$\ds x \in T \land y \in T$

Definition of $\RR$

$\ds $

$\leadsto$

$\ds y \in T \land x \in T$

Rule of Commutation

$\ds $

$\leadsto$

$\ds \tuple {y, x} \in \RR$

Definition of $\RR$

$\Box$

Transitive

$\RR$ is transitive:

$\ds x \in T \land y \in T$

$\leadstoandfrom$

$\ds \tuple {x, y} \in \RR$

Definition of $\RR$

$\ds y \in T \land z \in T$

$\leadstoandfrom$

$\ds \tuple {y, z} \in \RR$

Definition of $\RR$

$\ds \paren {x \in T \land y \in T} \land \paren {y \in T \land z \in T}$

$\leadstoandfrom$

$\ds \tuple {x, z} \in \RR$

Definition of $\RR$

$\Box$

So $\RR$ is reflexive, symmetric and transitive.

Hence, by definition, $\RR$ is an equivalence relation.

$\Box$

Test for Uniqueness

Now by definition of a partition, we have that:

$\mathbb S$ partitions $S \implies \forall x \in S: \exists T \in \mathbb S: x \in T$

Also:

$\ds x, y \in T$

$\leadsto$

$\ds \tuple {x, y} \in \RR$

Definition of $\RR$

$\ds x \in T_1, y \in T_2, T_1 \ne T_2$

$\leadsto$

$\ds \tuple{x, y} \notin \RR$

Definition of Set Partition

Thus $\mathbb S$ is the set of $\RR$-classes constructed above, and no other relation can be constructed in this way.

$\blacksquare$

Also see

Fundamental Theorem on Equivalence Relations for the converse.

Sources

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