Namespaces
Variants
Actions

Difference between revisions of "Kernel of a function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Start article: Kernel of a function)
 
(→‎References: isbn link)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
The  [[equivalence relation]] on the domain of the function expressing the  property that equivalent elements have the same image under the  function.
+
{{TEX|done}}{{MSC|03E}}
  
If $f : X \rightarrow Y$ then we define the relation $\stackrel{f}{\equiv}$ by
+
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) \ .  
 
x_1 \stackrel{f}{\equiv} x_2 \Leftrightarrow f(x_1) = f(x_2) \ .  
Line 7: Line 9:
 
The [[equivalence class]]es of $\stackrel{f}{\equiv}$ are the '''fibres''' of ''f''.
 
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
+
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 \, ,
 
q_\sim : x \mapsto [x]_\sim \, ,
 
$$
 
$$
where  $[x]_\sim\,$ is the equivalence class of $c$ 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====
 +
* Paul M. Cohn, ''Universal algebra'', Kluwer (1981) {{ISBN|90-277-1213-1}}

Latest revision as of 17:00, 23 November 2023

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=30497