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: $$[X]/R =\{S|(\exists x)(x\in X\land X\in S=[x]/R)\}=\{[x]/R|x\in X\}$$
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 $$X/C = \{(x,y) | (\exists S \in C)(x \exists \in S \land y \exists \in S)\}$$