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

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    There is no question mark in this post.2012-11-16
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    I'm sorry, but some of the definition formulas are all wonky and don't seem to make sense to me (namely, the formulas for $[X]/R$ and $X/C$.). They seem to contain mistakes.2012-11-16
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    Are you asking for clarification of the definitions, i.e., help to put into words what the definitions mean? Are you asking for what the definitions mean, intuitively?2012-11-16
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    Im just having issues putting the definitions into words2012-11-16
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    @mike4ty4 Kristen took the time to format as best as possible; I think some of the info got lost in the process.2012-11-16
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    @Kristen: So do you want the formulas, or a statement of the definition in words (i.e. a "translation" of the formulas to English words)?2012-11-16
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    Do you understand my comment above? Note: Every partition of a set determines an equivalence relation on that set, and every equivalence class induces a partition of the set into equivalence classes.2012-11-16
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    @amWhy: I suppose. But without clear formulas, it's not so easy to give an answer.2012-11-16
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    I assume Def 1 should read something like $[X]/R=\bigl\{S\subset X\mid \exists x\in X\colon S=\{y\in X\mid (x,y)\in R\}\bigr\}$ or $[X]/R =\bigl\{\{y\in X\mid (x,y)\in R\}\mid x\in X\bigr\}$, which are still monstrous, but seem to be correcter. And Def 3 should probably read something like $X/C=\{(x,y)\in X^2\mid \exists S\in C\colon x\in S\land y\in S\} $. (All give or take variations e.g. in quantor notation).2012-11-16
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    @Hagen von Eitzen: Yeah, that makes more sense.2012-11-16
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    these are the definitions I was given in class2012-11-16
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    @Kristen: Were you given them as printed notes or did you copy them from the board? I have very often seen students 'copy' all sorts of things from the board which the lecturer definitely didn't write!2012-11-16
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    Kristen: Are things clearing up for you, given the answers, or are you simply more confused?2012-11-16
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    The definitions are in a book that was written by my teacher...and I think I understand the partition a little more but the other concepts are still confusing me2012-11-16

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