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### 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 $x/\sim \ = \ y/\sim$
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Thu, 04 Jun 2020 13:24 GMT