A question in my book, chapter relations Let $f$ : $M \rightarrow N$ and $x R y \leftrightarrow f(x) = f(y)$ prove that this is an equivalence relation (the proof for it being an equivalence relation is pretty straight forward and easy thus already done), and for a $f$ : $M \rightarrow N$ injective, I should write the partition on $M$ Which is defined by $R$.
So it is the second part that I have problems with, how could I write this partition? What would the equivalence classes be?
Consider any $x \in M$. What is $x$ related to?
Well, under $R$, $xRy$ (for some $y \in M$) whenever $f(x) = f(y)$. However, $f$ is injective by hypothesis. Hence, $x=y$.
What this means is that each element is related only to itself.
Hence, the collection of equivalence classes is
$$M/R = \Big\{ \{x\} \,\Big|\, x \in M \Big\}$$