Difference between revisions of "Kernel of a function"
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| − | 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. |
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| + | 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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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 | + | 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 |
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q_\sim : x \mapsto [x]_\sim \, , | q_\sim : x \mapsto [x]_\sim \, , | ||
$$ | $$ | ||
| − | where $[x]_\sim\,$ is the equivalence class of $ | + | 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]]. |
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| + | See [[kernel congruence]] for the corresponding definition when the map is a [[homomorphism]] between [[algebraic system]]s. | ||
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| + | ====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
Kernel of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_function&oldid=30497