# Union of Equivalence Classes is Whole Set

## Theorem

Let $\RR \subseteq S \times S$ be an equivalence on a set $S$.

Then the set of $\RR$-classes constitutes the whole of $S$.

## Proof

We have that:

 $\, \ds \forall x \in S: \,$ $\ds x$ $\in$ $\ds \eqclass x \RR$ Definition of Equivalence Class $\text {(1)}: \quad$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \set {y \in S: x \mathrel \RR y}$ Definition of Equivalence Class

and:

 $\ds \eqclass x \RR$ $=$ $\ds \set {y: x \mathrel \RR y}$ Definition of Equivalence Class $\text {(2)}: \quad$ $\ds$ $\subseteq$ $\ds S$ Definition of Subset

Then:

 $\ds S$ $=$ $\ds \bigcup_{x \mathop \in S} \set x$ Definition of Union of Set of Sets $\ds$ $\subseteq$ $\ds \bigcup_{x \mathop \in S} \eqclass x \RR$ from $(1)$ $\ds$ $\subseteq$ $\ds \bigcup_{x \mathop \in S} S$ from $(2)$ $\ds$ $=$ $\ds S$

$\blacksquare$