For a set $S$ with an equivalence relation $R$, the set of equivalence classes is a partition of $S$ called the "quotient set of $S$ by $R$" and is denoted by $S/R$.
Now, let $f$ be a function defined on $S$. Consider the equivalence relation "$f$ maps to the same value". Is there a standard notation for the induced partition of $S$? For example, is it common to say "quotient set of $S$ by $f$" and denote by $S/f$?
The equivalence relation you mention is usually indicated by $\operatorname{ker}f$, so it follows that the induced quotient of S is indicated by: $$S/\operatorname{ker}f$$