Relation Partitions Set iff Equivalence/Proof

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 $6.4$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.