Relation Partitions Set iff Equivalence

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

  • 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 3$: Equivalence relations
  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations: Theorem $5$
  • 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Theorem $1.11$
  • 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
  • 2002: Thomas Jech: Set Theory (3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set
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