Equivalence Class Equivalent Statements

Theorem

Let $\mathcal R$ be an equivalence on $S$.

Then $\forall x, y \in S$, the following statements are all equivalent:

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

$(2): \quad \eqclass x {\mathcal R} = \eqclass y {\mathcal R}$

$(3): \quad x \mathrel {\mathcal R} y$

$(4): \quad x \in \eqclass y {\mathcal R}$

$(5): \quad y \in \eqclass x {\mathcal R}$

$(6): \quad \eqclass x {\mathcal R} \cap \eqclass y {\mathcal R} \ne \O$

Proof

$(1)$ and $(2)$ are equivalent because, by Equivalence Class is Unique, $\eqclass x {\mathcal R}$ is the unique $\mathcal R$-class to which $x$ belongs, and $\eqclass y {\mathcal R}$ is the unique $\mathcal R$-class to which $y$ belongs. As these are unique for each, they must therefore be the same set.

$(2)$ is equivalent to $(3)$ by Equivalence Class holds Equivalent Elements.

$(3)$ is equivalent to $(4)$ by the definition of Equivalence Class.

$(3)$ is equivalent to $(5)$ through dint of the symmetry of $\mathcal R$ and the definition of Equivalence Class.

$(3)$ is equivalent to $(6)$ from Equivalence Classes are Disjoint.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 10$: Theorem $10.4$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 17.7$: Equivalence classes

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups: $\text{I}$: Theorem $4$