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 ${\displaystyle f:X\rightarrow Y}$ then we define the relation ${\displaystyle {\stackrel {f}{\equiv }}}$ by

${\displaystyle x_{1}{\stackrel {f}{\equiv }}x_{2}\Leftrightarrow f(x_{1})=f(x_{2}).\,}$

The equivalence classes of ${\displaystyle {\stackrel {f}{\equiv }}}$ are the fibres of f.

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

${\displaystyle q_{\sim }:x\mapsto [x]_{\sim },\,}$

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