# Equivalnce relation proof

1. Mar 24, 2012

### Tokenfreak

I am trying to prove this as I am practicing for a test but I am pretty much clueless on this problem:

Prove that if ~ is an equivalence relation on a set s and [a] denotes the equivalence class of a in s under ~, then a ~ b if and only if [a] = .

If anyone can give me some points on how to approach or start this problem it would be great. Thanks.

2. Mar 24, 2012

### SteveL27

It falls directly out of the definition of equivalence relation, so it's tricky to think of a hint. But what happens if [a] $\neq$ ? Then there must either be an element in ____ that's not in ______ or vice versa. Then what?

3. Mar 25, 2012

### Tokenfreak

So would it be safe to assume that an equivalence relation ~ on a set s is a relation satisfying a,b in s. If [a] != , then there must be an element in a that's not in b or vice versa. Therefore, this contradicts that [a] = .

Is this close? I am pretty much clueless on this proof.

4. Mar 25, 2012

### SteveL27

No. You need to go back to your book and read what an equivalence relation is.

Aren't a and b assumed to be elements of s? So a,b in s is true of all a and b in s. Has nothing to do with equivalence relations.

Well yes, if you assume [a] != then that contradicts [a] = . Isn't that always the case no matter what [a] and are?

I think you need to read your text and/or class notes to understand what an equivalence relation is.

Last edited: Mar 25, 2012