The kernel of a function $f\colon A\to B$ is the equivalence relation $\mathord {\sim }_{\mathrm{Ker}\webleft (f\webright )}$ on $A$ obtained by declaring $a\sim _{\mathrm{Ker}\webleft (f\webright )}b$ iff $f\webleft (a\webright )=f\webleft (b\webright )$.[1]
Footnotes
[1] The kernel $\mathrm{Ker}\webleft (f\webright )\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ of $f$ is the underlying functor of the monad induced by the adjunction $\text{Gr}\webleft (f\webright )\dashv f^{-1}\colon A\mathbin {\rightleftarrows }B$ in $\textbf{Rel}$ of Chapter 6: Constructions With Relations, Item 2 of Proposition 6.3.1.1.2.
