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{{TEX|done}}{{MSC|03E}}

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 class]]es 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 system]]s.

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