Namespaces
Variants
Actions

Kernel of a function

From Encyclopedia of Mathematics
Revision as of 17:00, 23 November 2023 by Chapoton (talk | contribs) (→‎References: isbn link)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

2020 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://encyclopediaofmath.org/index.php?title=Kernel_of_a_function&oldid=54627