Given a function $f : A → B$, let $R$ be the relation defined on $A$ by $aRa′$ whenever $f(a) = f(a′)$. Prove that $R$ is an equivalence relation and determine the equivalence classes.
To prove that $R$ is an equivalence relation, I know I have to show that R is reflexive, symmetric, and transitive. And from there, I can also determine the equivalence classes. However, I'm not sure where exactly to start. What exactly is the relation?
On
It is equivalent to speak about an equivalence relation $R$ on a set $A$ and a partition of $A$ into its classes of equivalence.
Hence there is a natural partition by the preimages $f^{-1}(b)$ for all $b \in B$ (which are either disjoint or identical), in other terms:
$$aRa' \Leftrightarrow f(a)=f(a') \Leftrightarrow a \ \text{and} \ a' \ \text{belong to the same preimage} $$
$f(a) = f(a) \implies R$ is reflexive, and if $aRb \implies f(a) = f(b) \implies f(b) = f(a) \implies bRa$, hence $R$ is symmetric. And if $aRb, bRc \implies f(a) = f(b), f(b) = f(c) \implies f(a) = f(c) \implies aRc$. Thus $R$ is equivalence relation on $A$. The equivalent classes consist of the $f^{-1}(x)'s $ whereas $x \in $ the range of $f$.