File:Largest subset on which a function exists to complete the triangle.pdf
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Largest_subset_on_which_a_function_exists_to_complete_the_triangle.pdf (308 × 139 pixels, file size: 39 KB, MIME type: application/pdf)
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D_F is the largest subset of Y on which it is possible to unambiguously define a function <math>f : D_F \to Z</math> that satisfies <math>F = f \circ \pi.</math> That is, it turns the diagram on the left into the triangular diagram on the right.
Summary[edit]
| Description |
English: The map is a surjection is a surjection. In the diagram on the left, there might not exist any function that when added to the (left) diagram, will make it commute; that is, there might not exist any such that on This motivates defining the set which is defined as the largest subset of on which it is possible to unambiguously define a function that satisfies on
In the diagram on the right, the arrows labelled and are actually the restrictions to of the original maps and That is, it is the largest subset of on which a edge may be added to the diagram on the left to turn it into the triangular diagram on the right (after restricting the functions to the appropriate subsets). |
| Date | |
| Source | Own work |
| Author | Mgkrupa |
LaTeX Source Code[edit]
%Note: Add the -shell-escape flag to the compilation command to compile directly to an image.
%For .pdf files, use \documentclass[preview]{standalone}
%For .png files, (see http://tex.stackexchange.com/q/11866 for more info) use: \documentclass[convert={density=300,size=1080x800,outext=.png}]{standalone}
%For .svg files you need pdf2svg installed (see https://tex.stackexchange.com/a/51766/187911 for more info) use:
\documentclass[convert={outext=.svg,command=\unexpanded{pdf2svg \infile\space\outfile}},multi=false]{standalone}
%If you get an error compiling with this then it might be necessary to initially compile this code into a .pdf file one time (to do this, replace the \documentclass[]{} code above with \documentclass[preview]{standalone} and then compile; afterwards, undo this code replacement). This explicit compilation into a .pdf file needs to be done ONLY once. Afterwards, you should then be able to compile the .svg file using the above code.
\usepackage{tikz}
\usetikzlibrary{matrix,arrows,decorations.pathmorphing}
\usepackage[all]{xy}
\begin{document}
\begin{tikzpicture}[scale=1.5]
\node (X) at (0,1) {$X$};
\node (Y) at (0,0) {$Y$};
\node (Z) at (1,0) {$Z$};
\node (XD) at (2,1) {$\pi^{-1}\left(D_F\right)$};
\node (D) at (2,0) {$D_F$};
\node (Z2) at (3,0) {$Z$};
\path[->,font=\scriptsize,>=angle 90]
(X) edge node[left]{\scalebox{1.5}{$\pi$}} (Y)
(X) edge node[above right]{\scalebox{1.5}{$F$}} (Z)
(XD) edge node[left]{\scalebox{1.5}{$\pi$}} (D)
(XD) edge node[above right]{\scalebox{1.5}{$F$}} (Z2)
(D) edge[dashed] node[below]{\scalebox{1.5}{$f$}} (Z2);
\end{tikzpicture}
\end{document}
Licensing[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 20:38, 25 April 2021 |  | 308 × 139 (39 KB) | Mgkrupa (talk | contribs) | Uploaded own work with UploadWizard |
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