# Definition:Equivalence Relation Induced by Partition

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## Definition

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:

- $\Bbb S = S / \RR$

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 7$: Relations - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: Problem Set $\text{A}.3$: $16$