Element in its own Equivalence Class

Theorem

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

Then every element of $S$ is in its own $\RR$-class:

$\forall x \in S: x \in \eqclass x \RR$

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

$\ds \tuple {x, x}$

$\in$

$\ds \RR$

Definition of Equivalence Relation: $\RR$ is Reflexive

$\ds \leadsto \ \ $

$\ds x$

$\in$

$\ds \eqclass x \RR$

Definition of Equivalence Class

$\blacksquare$

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): $\S 17.1$: Equivalence classes