Definitions

6. Equivalence class and X modulo $\sim$

$$ \forall x \in X: x/\sim \ = {\ y \in X\ |\ y \sim x } \ X/\sim \ ={x/\sim \ |\ x \in X } $$

7.

$P$ is a partition of non-empty set $X$. The relation $X/P$ is a relation on $X$. $x(X/P)y$(or $x,y \in X/P$) iff $\exists A \in P$ so that $x, y \in A$.

Theorems

3.

Let $\sim$ be an equivalence relation on the non empty set $X$.

1. Any $x/\sim$ is a non empty subset of $X$.

2. $x \sim y$ iff $x/\sim \ \cap \ y/\sim \ \neq \ \emptyset$

3. $x \sim y$ iff $x/\sim \ = \ y/\sim$