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 $\text{A}.3$: $16$