Equivalence Class is not Empty

Theorem

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

Then no $\RR$-class is empty.

Proof

\(\ds \forall \eqclass x \RR \subseteq S: \exists x \in S: \, \)

\(\ds x\)

\(\in\)

\(\ds \eqclass x \RR\)

Definition of Equivalence Class

\(\ds \leadsto \ \ \)

\(\ds \eqclass x \RR\)

\(\ne\)

\(\ds \O\)

Definition of Empty Set

$\blacksquare$

Also see

• Fundamental Theorem on Equivalence Relations

• Union of Equivalence Classes is Whole Set
• Equivalence Classes are Disjoint

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations
• 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): $\S 17.5 \ \text{(iii)}$: Equivalence classes
• 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: Exercises: $1.8$