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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.

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