Equivalence Class Equivalent Statements

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

Let $x, y \in S$.


 * $(1): \quad x$ and $y$ are in the same $\RR$-class
 * $(2): \quad \eqclass x \RR = \eqclass y \RR$
 * $(3): \quad x \mathrel \RR y$
 * $(4): \quad x \in \eqclass y \RR$
 * $(5): \quad y \in \eqclass x \RR$
 * $(6): \quad \eqclass x \RR \cap \eqclass y \RR \ne \O$

$(1)$ Equivalent to $(2)$

 * $(1)$ and $(2)$ are equivalent because, 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 therefore be the same set.

$(2)$ Equivalent to $(3)$

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

$(3)$ Equivalent to $(4)$

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

$(3)$ Equivalent to $(5)$

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

$(3)$ Equivalent to $(6)$

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