How Does Equivalence Class Equality Work in Set Theory?

[quasar_4]

Homework Statement

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]]=[].

Homework Equations

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

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.

 

[matt grime]

You can't write [[a]]Rb. The left hand side is an equivalence class, and the RHS is an element. ?R? only makes sense if both sides are in S. You also at one point say that b is an element of S and b is an equivalence relation.

To show that []=[[a]] or indeed any set eqaulity all you need to do is demonstrate that if x is [] it is in [[a]], and if y is in [[a]] it is in []. These are immediate from the definitions, and the hypothesis that b is in [[a]]. For we want to take x in [], i.e. xRb, and show that this implies x is in [[a]], i.e. that xRa. So, if xRb, and bRa, then does xRa?

 

[HallsofIvy]

Homework Statement

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]]=[].

Homework Equations

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

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.

You mean 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.

Again, no. [[a]] is a subset of A, not a member. You mean: 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]] = [].

No, b is definitely NOT an equivalence relation! b is a member of set A and an equivalence relation is a subset of Rx R.

[/quote]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. [/QUOTE]

You want to show that [[a]]= [] which means you want to show two sets are equal. You do that by showing any member of one is a member of the other. If c is in [[a]], then c R a. Since you are told that b is in [[a]], b R a also. What does that tell you about c and b? Going the other way, while you were told that b is in [[a]], you are NOT told directly that a is in []. You will have to prove that.