Equivalence Class Equivalent Statements/1 iff 2

Theorem

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

Let $x, y \in S$.

The following statements are equivalent:

$(1): \quad x$ and $y$ are in the same $\RR$-class

$(2): \quad \eqclass x \RR = \eqclass y \RR$

Proof

By Equivalence Class is Unique:

$\eqclass x \RR$ is the unique $\RR$-class to which $x$ belongs

and:

$\eqclass y \RR$ is the unique $\RR$-class to which $y$ belongs.

As these are unique for each, they must be the same set.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 17.7$: Equivalence classes