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 <math>f : X \rightarrow Y</math> then we define the relation <math>\stackrel{f}{\equiv}</math> by

<math>x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) . \,</math>

The equivalence classes of <math>\stackrel{f}{\equiv}</math> are the fibres of f.

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

<math>q_\sim : x \mapsto [x]_\sim , \, </math>

where <math>[x]_\sim\,</math> is the equivalence class of x under <math>\sim\,</math>. Then the kernel of the quotient map <math>q_\sim\,</math> is just <math>\sim\,</math>. This may be regarded as the set-theoretic version of the First Isomorphism Theorem.