# 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  then we define the relation  by



The equivalence classes of  are the fibres of f.

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



where  is the equivalence class of x under . Then the kernel of the quotient map  is just . This may be regarded as the set-theoretic version of the First Isomorphism Theorem.