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 $\sim \,$. This may be regarded as the set-theoretic version of the First Isomorphism Theorem.