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Difference between revisions of "Kernel of a function"

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(References: Cohn (2003))
(See also kernel congruence)
 
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The  [[equivalence relation]] on the domain of a function expressing the  property that equivalent elements have the same image under the  function.
 
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 $\stackrel{f}{\equiv}$ by
+
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) \ .  
 
x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) \ .  
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$$
 
$$
 
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]].
 
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====
 
====References====
 
* Paul M. Cohn, ''Universal algebra'', Kluwer (1981) ISBN 90-277-1213-1
 
* Paul M. Cohn, ''Universal algebra'', Kluwer (1981) ISBN 90-277-1213-1

Latest revision as of 07:36, 13 November 2016

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://encyclopediaofmath.org/index.php?title=Kernel_of_a_function&oldid=39726