(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove: If E_{1}, · · · , E_{k}are the disjoint equivalence classes

determined by an equivalence relation R over a set X, then

(a) X = union of disjoint equivalence classes E_{j}

(b) R = union of disjoint (E_{j}x E_{j})

2. Relevant equations

R is a subset of X_{x}X

3. The attempt at a solution

For (a), my thoughts are :

1/ By reflexive property of equiv. relation, there exists an element a in X such that <a,a> belongs to R

2/ I know "E_{1}, · · · , E_{k}are the disjoint equivalence classes

determined by an equivalence relation R", so if a belongs to R, then a must also belongs to some of the equivalence classes.

3/ Then I use the fact that a is in X, and a belongs to some of equivalence classes, then X must be the union of those equiv. classes

But then I'm not sure if my thoughts are correct, probably what I'm confused is with my 3rd idea.

For (b), my thoughts are:

Since R is a subset of X x X

and by (a), X is an union of disjoint equiv. classes

then X x X = union of (E_{j}x E_{j})

And again, I feel shaky about my reasoning

Would someone please give me some hints or ideas?

I really appreciate your time and your help.

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# Prove a set X is union of disjoint equivalence classes

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