Definition:Equivalence Relation Induced by Partition

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Let $S$ be a set.

Let $\Bbb S$ be a partition of $S$.

Let $\RR \subseteq S \times S$ be the relation defined as:

$\forall \tuple {x, y} \in S \times S: \tuple {x, y} \in \RR \iff \exists T \in \Bbb S: \set {x, y} \subseteq T$

Then $\RR$ is the (equivalence) relation induced by (the partition) $\Bbb S$.

Also known as

Some sources refer to this as the (equivalence) relation defined by (the partition) $\Bbb S$.

Also see

It is proved in Relation Induced by Partition is Equivalence that:

$\RR$ is unique
$\RR$ is an equivalence relation on $S$.

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

$\Bbb S = S / \RR$