Proof of equivalence relation on set.

Question

I'm new to the whole relations topic and stumbled upon a problem.

I know that an equivalence relation is a relation that is symmetric, transitive, reflexive, (and not usually anti-symmetric).

But then how do we prove that for a equivalence relation R on a set X, and if a,b ∈ X,

1. a∈ [[a]] <--equivalence class
2. [[a]] = [[b]] iff (a,b) ∈ R
3. [[a]] ∩ [[b]] = 0 iff (a,b) ∉ R

?

Answer 1

• $a\in [[a]]$ is easy to prove. You can see that since $aRa$, $a\in [[a]]$.

• $[[a]] = [[b]]$ iff $(a,b) ∈ R$ is easy to prove. If $(a,b)\in R$, then $[[a]]=[[b]]$. If $[[a]]=[[b]]$, then $aRb$, i.e., $(a,b)\in R$.

• $[[a]]\cap [[b]]=\emptyset$ iff $(a,b)\not\in R$. If $(a,b)\not\in R$, then $[[a]]\cap[[b]]=\emptyset$ by definition. If $[[a]]\neq[[b]]$, then $a\not R b$, i.e., $[[a]]\cap[[b]]=\emptyset$.