Definition:Relation Induced by Partition

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Definition

Let $S$ be a set.

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

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

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


Then $\mathcal R$ 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:

$\mathcal R$ is unique
$\mathcal R$ is an equivalence relation on $S$.

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

$\Bbb S = S / \mathcal R$


Sources