Prove a set X is union of disjoint equivalence classes

[Ceci020]

Homework Statement

Prove: If E₁, · · · , 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)

Homework Equations

R is a subset of X _x X

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 "E₁, · · · , 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.

 

[Bacle2]

Hi:
You know an equivalence relation partitions your set into disjoint equivalence classes,right? If aRb and aRc, then bRa (symmetry) and aRc , so bRc (so b and c are in the same class) , so no element belongs to more than one equivalent class, and all elements belong to some class.

For (b), don't you mean R=EixEj , for Ei,Ej disjoint?