# Kernel of a function

2010 Mathematics Subject Classification: *Primary:* 03E [MSN][ZBL]

The equivalence relation on the domain of a function expressing the property that equivalent elements have the same image under the function.

If $f : X \rightarrow Y$ then we define the relation $\stackrel{f}{\equiv}$ on $X$ 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.

See kernel congruence for the corresponding definition when the map is a homomorphism between algebraic systems.

#### References

- Paul M. Cohn,
*Universal algebra*, Kluwer (1981) ISBN 90-277-1213-1

**How to Cite This Entry:**

Kernel of a function.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Kernel_of_a_function&oldid=39726