Equivalence Relation

Last updated Nov 1, 2022

Let $X$ be a Set and let $R \subset X \times X$ be a Relation on $X$. Then $R$ is an Equivalence Relation if it is a Preorder Relation with the following additional property:

• Symmetry: If $x, y \in X$ and $xRy$, then $yRx$.

1. As with Preorders, $\pi_{1}(R) = \pi_{2}(R) = X$.
2. We often denote $R$ by $\sim$ or $=$.
3. If $(x, y) \not\in R$, then we write $x \not\sim y$ or $x \neq y$.
4. Correspondence of Surjective Functions, Partitions, and Equivalence Relations