Union of Equivalence Classes is Whole Set

Theorem

Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.

Then the set of $\RR$-classes constitutes the whole of $S$.

We have that:

$\ds \forall x \in S: \, $

$\ds x$

$\in$

$\ds \eqclass x \RR$

Definition of Equivalence Class

$\text {(1)}: \quad$

$\ds \leadsto \ \ $

$\ds x$

$\in$

$\ds \set {y \in S: x \mathrel \RR y}$

Definition of Equivalence Class

and:

$\ds \eqclass x \RR$

$=$

$\ds \set {y: x \mathrel \RR y}$

Definition of Equivalence Class

$\text {(2)}: \quad$

$\ds $

$\subseteq$

$\ds S$

Definition of Subset

Then:

$\ds S$

$=$

$\ds \bigcup_{x \mathop \in S} \set x$

Definition of Union of Set of Sets

$\ds $

$\subseteq$

$\ds \bigcup_{x \mathop \in S} \eqclass x \RR$

$\ds $

$\subseteq$

$\ds \bigcup_{x \mathop \in S} S$

$\ds $

$=$

$\ds S$

$\blacksquare$

1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations: Theorem $\text{A}.2$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 17.5 \ \text{(i)}$: Equivalence classes
1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: Exercises: $1.8$
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations

Greg Martin (https://math.stackexchange.com/users/16078/greg-martin), Prove union of equivalence classes is the whole set, URL (version: 2020-01-30): https://math.stackexchange.com/q/3527923