TITLE: Understanding equivalence class, equivalence relation, partition
QUESTION [21 upvotes]: I'm having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions.
1) The collection of equivalence classes w.r.t. $R$

Def: Let $R$ be an equivalence relation in a set $X$. The collection of equivalence classes w.r.t. $R$ is the set:
  $$[X]/R =\{S|(\exists x)(x\in X\land S=[x]/R)\}=\{[x]/R|x\in X\}$$

2) Partition

Def: Let $X$ be a set. A collection of sets $C$ is a partition of $X$ if:
  (i) $$\bigcup_{S\in\ C} S=X.$$
  (ii) $$\forall S \in C, S \neq \varnothing$$
  (iii) $$\forall S, S' \in C, S' \neq S \Rightarrow S \cap S' = \varnothing$$

3) Relation induced by

Def: Let $C$ be a partition of $X$. The relation induced by $C$, denoted by $X/C$, is a relation in $X$ such that $$X/C = \{(x,y) | (\exists S \in C)(\exists x \in S \land \exists y \in S)\}$$

REPLY [2 votes]: The definitions of the concepts in words are

The equivalence classes of an equivalence relation are subsets of the set on which the equivalence relation is defined, such that two elements are in the same equivalence class if and only if they are related by the equivalence relation.
A partition of a set is a collection of subsets of the set, whose union is the whole set, and such that no two subsets share any elements in common, and none of which are the empty set.
The relation induced by a partition is the relation given by "the two objects are related if they lie in the same cell of the partition" (where a "cell" is one of the subsets of the partitioned set that make up the partition.).