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:

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

Proof

\(\ds \forall x \in S: \, \)

\(\ds \tuple {x, x}\)

\(\in\)

\(\ds \RR\)

Definition of Equivalence Relation: $\RR$ is Reflexive

\(\ds \leadsto \ \ \)

\(\ds x\)

\(\in\)

\(\ds \eqclass x \RR\)

Definition of Equivalence Class

$\blacksquare$

Sources

• 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