I have defined an equivalence relation $R$ such that $a R b \iff f(a) = f(b)$, where $f: A \to B$ is a surjective function. How can I prove that $R$ is an equivalence relation?. I know it has to be reflexive, symmetric and transitive. Here is what I have tried:
- Reflexive: $\forall a \in A$ it is clear that $f(a) = f(a)$. Thus, $aRa$.
- Symmetric: Let $a,b \in A$ s.t. $aRb$, then $f(a)=f(b)$, which also implies that $f(b)=f(a)$. Therefore, $bRa$.
- Transitive: Let $a,b,c\in A$ s.t $aRb$ and $bRc \iff f(a)=f(b)$ and $f(b) = f(c)\implies f(a)=f(c)$. This fact again implies that $aRc$.
Is this correct? Do $a,b,c$ belong to $A$ or I just misunderstood where they truly belong?