Relation Partitions Set iff Equivalence/Proof
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Theorem
Let $\RR$ be a relation on a set $S$.
Then $S$ can be partitioned into subsets by $\RR$ if and only if $\RR$ is an equivalence relation on $S$.
The partition of $S$ defined by $\RR$ is the quotient set $S / \RR$.
Proof
Let $\RR$ be an equivalence relation on $S$.
From the Fundamental Theorem on Equivalence Relations, we have that the equivalence classes of $\RR$ form a partition.
$\Box$
Let $S$ be partitioned into subsets by a relation $\RR$.
From Relation Induced by Partition is Equivalence, $\RR$ is an equivalence relation.
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $6$
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.3$: Theorem $5$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{E}$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations: Theorem $4$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.4$