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

Prove the following statement:

Let R be an equivalence relation on set A. If b is in the equivalence class of a, denoted [[a]] then [[a]]=[].

2. Relevant equations

[[a]], [[a]]=[]; definition of equivalence: a relation R on a set A that is reflexive, symmetric and transitive is an equivalence relation.

3. The attempt at a solution

Consider an element b in set {x in S| x R a} denoted by the equivalence relation [[a]]. If b is in this set, it is an element of the equivalence class. It follows that if b is an element of [[a]], then it must be an equivalence relation and is reflexive such that [[a]] R b. It is also symmetric, such that [[a]] R b = b R [[a]], and transitive such that for any c in [[a]], if [[a]] R b and b R c then [[a]] R c. Thus, b is also an equivalence relation and the set {x in S|x R b} may be denoted by []. Hence, [[a]] = [].

I'm not sure if I was on track here or not. I feel as if I went in a circle without actually proving anything. I'm not sure exactely what else to do, but I thought maybe to show they are equal, I have to express the sets as equal somehow. Or would I go about it by saying that [[a]] = [] if a = b? Any critique of this proof is highly welcome -- I'm pretty new at this whole thing.

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# Equivalence relation proof

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