Kernel of a function

In set theory, the kernel of a function is the equivalence relation on the domain of the function expressing the property that equivalent elements have the same image under the function.

If $$f : X \rightarrow Y$$ then we define the relation $$\stackrel{f}{\equiv}$$ by


 * $$x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) . \,$$

The equivalence classes of $$\stackrel{f}{\equiv}$$ are the fibres of f.

Every function gives rise to an equivalence relation as kernel. Conversely, every equivalence relation $$\sim\,$$ on a set X gives rise to a function of which it is the kernel. Consider the quotient set $$X/\sim\,$$ of equivalence classes under $$\sim\,$$ and consider the quotient map $$q_\sim : X \rightarrow X/\sim$$ defined by


 * $$q_\sim : x \mapsto [x]_\sim, \, $$

where $$[x]_\sim\,$$ is the equivalence class of x under $$\sim\,$$. Then the kernel of the quotient map $$q_\sim$$ is just $$\sim\,$$.