Citizendium - building a quality free general knowledge encyclopedia. Click here to join and contribute—free Many thanks December donors; special to Darren Duncan. January donations open; need minimum total \$100. Let's exceed that. Donate here. By donating you gift yourself and CZ.

# Kernel of a function

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

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.