Kernel of a function

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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.