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) $$x$$ and $$y$$ are in the same $\mathcal{R}$-class;
 * 2) $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}} = \left[\!\left[{y}\right]\!\right]_{\mathcal{R}}$$;
 * 3) $$x \mathcal{R}y$$;
 * 4) $$x \in \left[\!\left[{y}\right]\!\right]_{\mathcal{R}}$$;
 * 5) $$y \in \left[\!\left[{x}\right]\!\right]_{\mathcal{R}}$$;
 * 6) $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}} \cap \left[\!\left[{x}\right]\!\right]_{\mathcal{R}} \ne \varnothing$$.

Proof

 * 1 and 2 are equivalent because, by One and Only Equivalence Class, $$\left[\!\left[{x}\right]\!\right]_{\mathcal{R}}$$ is the unique $\mathcal{R}$-class to which $$x$$ belongs, and $$\left[\!\left[{y}\right]\!\right]_{\mathcal{R}}$$ 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.