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Passing to the quotient (function) - Revision history
2024-03-29T13:31:06Z
Revision history for this page on the wiki
MediaWiki 1.24.1
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=3286&oldid=prev
Alec: /* Proof of claims */ Added link to surjective proof
2016-10-11T20:49:19Z
<p><span dir="auto"><span class="autocomment">Proof of claims: </span> Added link to surjective proof</span></p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 20:49, 11 October 2016</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>*: So it is easy to see that we require {{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} in order to proceed.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>*: So it is easy to see that we require {{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} in order to proceed.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* To see that '''if {{M|f}} is surjective so is {{M|\tilde{f} }} see my notes here: </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">** [[Talk:Passing_to_the_quotient_(function)/Induced_is_surjective_iff_function_is_surjective]]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Requires proof|grade=A|msg=Most of the proofs are done, I've done the surjective one like 3 times (CHECK THE TALK PAGE! SO YOU DON'T DO IT A FOURTH!) Also:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Requires proof|grade=A|msg=Most of the proofs are done, I've done the surjective one like 3 times (CHECK THE TALK PAGE! SO YOU DON'T DO IT A FOURTH!) Also:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Move the proofs into sub-pages. It is just so much neater!}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Move the proofs into sub-pages. It is just so much neater!}}</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==See also==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Passing to the quotient]] - disambiguation page</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Passing to the quotient]] - disambiguation page</div></td></tr>
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Alec
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=3229&oldid=prev
Alec: Reclassified as theorem, cleaned up formatting, cleaned up writing style. Moved diagram into subpage, added explanation, corrected notation, basically redone it all!
2016-10-08T22:10:45Z
<p>Reclassified as theorem, cleaned up formatting, cleaned up writing style. Moved diagram into subpage, added explanation, corrected notation, basically redone it all!</p>
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<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 22:10, 8 October 2016</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==<del class="diffchange diffchange-inline">Definition</del>==</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{Refactor notice|review</ins>=<ins class="diffchange diffchange-inline">true}}</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}} <del class="diffchange diffchange-inline">(</del>I have chosen {{M|W}} to mean "whatever"<del class="diffchange diffchange-inline">) we can say:</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">: See [[Passing to the quotient]] for a disambiguation of this term.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">: '</del>''{{M|f}} may be factored through {{M|w}}'''</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">__TOC__</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">if </del>{{M|f}} <del class="diffchange diffchange-inline">and </del>{{M|w}} <del class="diffchange diffchange-inline">are such that</del>:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=<ins class="diffchange diffchange-inline">=Statement</ins>==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">* </del><math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math>  </div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{float-right|{{/Diagram}}|style=max-width:20em;}}</ins>Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}}<ins class="diffchange diffchange-inline"><ref group="Note"></ins>I have chosen {{M|W}} to mean "whatever"<ins class="diffchange diffchange-inline"></ref> then "</ins>''{{M|f}} may be factored through {{M|w}}''<ins class="diffchange diffchange-inline">" if<ref>Alec</ins>'<ins class="diffchange diffchange-inline">s own work, "distilled" from [[passing to the quotient (topology)]] which is defined by Mond (2013, Topology) and Lee (Intro to Top manifolds), by further abstracting the claim</ref>:</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">*: (this is the same as: </del><math>\forall x,y\in X[f(x)\ne f(y)\implies w(x)\ne w(y)]</math><del class="diffchange diffchange-inline">)</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* </ins>{{M|f}} <ins class="diffchange diffchange-inline">is constant on the {{plural|fibre|s}} of </ins>{{M|w}}<ins class="diffchange diffchange-inline"><ref group="Note">We can state this in 2 other equivalent ways</ins>:  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Then </del>{{M|f}} ''induces'' a <del class="diffchange diffchange-inline">function</del>, {{M|\tilde{f} }} such that <math>f=\tilde{f}\circ w</math>, <del class="diffchange diffchange-inline">or more simply that </del>the <del class="diffchange diffchange-inline">following </del>[[Commutative diagram|<del class="diffchange diffchange-inline">diagram </del>commutes]]<del class="diffchange diffchange-inline">:</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins><math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math>  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{| <del class="diffchange diffchange-inline">class</del>=<del class="diffchange diffchange-inline">"wikitable" border</del>="<del class="diffchange diffchange-inline">1</del>"</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># </ins><math>\forall x,y\in X[f(x)\ne f(y)\implies w(x)\ne w(y)]</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|<del class="diffchange diffchange-inline">-</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">See [[equivalent conditions to being constant on the fibres of a map]] for proofs and more details</ref><!--</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <del class="diffchange diffchange-inline">style="font</del>-<del class="diffchange diffchange-inline">size:</del>1<del class="diffchange diffchange-inline">.5em;" </del>| <del class="diffchange diffchange-inline"><math></del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">\begin</del>{<del class="diffchange diffchange-inline">xy</del>}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">END OF NOTE ON CONSTANT-ON-FIBRE PART</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del class="diffchange diffchange-inline">xymatrix</del>{</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">--></ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">X </del>\<del class="diffchange diffchange-inline">ar</del>[<del class="diffchange diffchange-inline">r]^w \ar</del>[<del class="diffchange diffchange-inline">dr</del>]<del class="diffchange diffchange-inline">_f & W \ar@</del>{<del class="diffchange diffchange-inline">.></del>}[<del class="diffchange diffchange-inline">d</del>]<del class="diffchange diffchange-inline">^</del>{\<del class="diffchange diffchange-inline">tilde</del>{f}}\\</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">If this condition is met then </ins>{{M|f}} ''induces'' a <ins class="diffchange diffchange-inline">[[mapping]]</ins>, {{M|\tilde{f}<ins class="diffchange diffchange-inline">:W\rightarrow Y </ins>}}<ins class="diffchange diffchange-inline">, </ins>such that <math>f=\tilde{f}\circ w</math> <ins class="diffchange diffchange-inline">(equivalently</ins>, the <ins class="diffchange diffchange-inline">diagram on the right </ins>[[Commutative diagram|commutes]]<ins class="diffchange diffchange-inline">). </ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"> & Y</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* </ins>{<ins class="diffchange diffchange-inline">{M</ins>|<ins class="diffchange diffchange-inline">\tilde{f}:W\rightarrow X}} may be given explicitly as: {{M|1</ins>=<ins class="diffchange diffchange-inline">\tilde{f}:v\mapsto f(w^{-1}(v))}}<ref group</ins>="<ins class="diffchange diffchange-inline">Note</ins>"<ins class="diffchange diffchange-inline">>Of course, only {{plural</ins>|<ins class="diffchange diffchange-inline">bijection</ins>|<ins class="diffchange diffchange-inline">s}} have {{plural|inverse function|s}}, we indulge in the common practice of using {{M|w^{</ins>-1<ins class="diffchange diffchange-inline">}(v)}} to mean {{M</ins>|<ins class="diffchange diffchange-inline">w^</ins>{<ins class="diffchange diffchange-inline">-1</ins>}<ins class="diffchange diffchange-inline">(</ins>\{<ins class="diffchange diffchange-inline">v</ins>\<ins class="diffchange diffchange-inline">})}}, in general for </ins>[[<ins class="diffchange diffchange-inline">sets]</ins>] {<ins class="diffchange diffchange-inline">{M|A</ins>}<ins class="diffchange diffchange-inline">} and {{M|B}} and a </ins>[<ins class="diffchange diffchange-inline">[mapping]</ins>] {<ins class="diffchange diffchange-inline">{M|f:A</ins>\<ins class="diffchange diffchange-inline">rightarrow B}} we use </ins>{<ins class="diffchange diffchange-inline">{M|</ins>f<ins class="diffchange diffchange-inline">^{-1</ins>}<ins class="diffchange diffchange-inline">(C)</ins>}<ins class="diffchange diffchange-inline">} to denote (for some {{M|C</ins>\<ins class="diffchange diffchange-inline">in</ins>\<ins class="diffchange diffchange-inline">mathcal{P</ins>}<ins class="diffchange diffchange-inline">(B)}} (a [[subset of]] </ins>{<ins class="diffchange diffchange-inline">{M|X</ins>}<ins class="diffchange diffchange-inline">})) the {{link</ins>|<ins class="diffchange diffchange-inline">pre</ins>-<ins class="diffchange diffchange-inline">image</ins>|<ins class="diffchange diffchange-inline">map</ins>}<ins class="diffchange diffchange-inline">} of </ins>{{M|<ins class="diffchange diffchange-inline">C}} under the [[function]] {{M|f}}, {{M|1=f^{-1}(C):=</ins>\{<ins class="diffchange diffchange-inline">a\in A\ \vert\ </ins>f<ins class="diffchange diffchange-inline">(a)\in C\</ins>} }}<ins class="diffchange diffchange-inline">. Just </ins>as <ins class="diffchange diffchange-inline">for </ins>{{M|<ins class="diffchange diffchange-inline">D</ins>\<ins class="diffchange diffchange-inline">in\mathcal</ins>{<ins class="diffchange diffchange-inline">P}(A)}} (a subset of {{M|A}}) we use {{M|</ins>f<ins class="diffchange diffchange-inline">(D)</ins>}<ins class="diffchange diffchange-inline">} to denote the {{link|image|function}} of {{M|D}} under {{M|f}}, namely</ins>: <ins class="diffchange diffchange-inline">{{M|1=f(D):=\{f(d)\in B\ \vert\ d\in D</ins>\} }<ins class="diffchange diffchange-inline">}</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{Warning box|1=Writing </ins>{{M|\tilde{f}:v\mapsto f(w^{-1}(v))}} <ins class="diffchange diffchange-inline">is dangerous as it may not be "''[[well-defined]]''"|2=A [[function]] (considered as a [[relation]]) of the form {{M|f:X\rightarrow Y}} must associate every {{M|x\in X}} with exactly one {{M|y\in Y}}.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">\end</del>{<del class="diffchange diffchange-inline">xy</del>}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Suppose that </ins>{{M|1=<ins class="diffchange diffchange-inline">w^{-1}(v)}} is [[empty-set|empty]] or contains 2 (or more!) elements, then what do we define {{M|</ins>\tilde{f} <ins class="diffchange diffchange-inline">}} as?</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">! Diagram</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">As it turns out it doesn't matter, but is really important to see why we must be so careful! This is why we require {{M|f}} to be constant on the fibres of {{M|w}}, as if we have {{M|1=w(x)=w(y)}} but {{M|f(x)\ne f(y)}} then no function composed with {{M|w}} can ever be equal to {{M|f}}!</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* Suppose that {{M|g</ins>:<ins class="diffchange diffchange-inline">W\rightarrow Y}} is such that {{M|1</ins>=f<ins class="diffchange diffchange-inline">=g</ins>\circ w<ins class="diffchange diffchange-inline">}}, then </ins>{<ins class="diffchange diffchange-inline">{M|</ins>1<ins class="diffchange diffchange-inline">=f(x)=g(w(x))</ins>}}<ins class="diffchange diffchange-inline">, and we have {{M|1=f(x)\ne f(y)</ins>}<ins class="diffchange diffchange-inline">}, then:</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">Note:</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*<ins class="diffchange diffchange-inline">* </ins>{{M|<ins class="diffchange diffchange-inline">1=</ins>w<ins class="diffchange diffchange-inline">(x)=w(y)}} so we must have </ins>{<ins class="diffchange diffchange-inline">{M|</ins>1<ins class="diffchange diffchange-inline">=g(w(x))=g(w(y))</ins>}<ins class="diffchange diffchange-inline">}, so we must have {{M|1=f</ins>(x<ins class="diffchange diffchange-inline">)=f(y</ins>)}}<ins class="diffchange diffchange-inline">! A contradiction!</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"># </del>{{M|\<del class="diffchange diffchange-inline">tilde</del>{f} }} <del class="diffchange diffchange-inline">may be explicitly written </del>as {{M|\<del class="diffchange diffchange-inline">tilde</del>{f}:<del class="diffchange diffchange-inline">W</del>\<del class="diffchange diffchange-inline">rightarrow Y</del>}} <del class="diffchange diffchange-inline">by </del>{{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Lastly note the alternate forms of the "constant on fibres" (in the note above) </ins>is <ins class="diffchange diffchange-inline">''very'' similar to the definition </ins>of <ins class="diffchange diffchange-inline">a function being [[injective]]</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">#* Or indeed </del>{{M|1=\tilde{f}:=f\circ w<del class="diffchange diffchange-inline">^</del>{<del class="diffchange diffchange-inline">-</del>1} }}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{{Todo|Develop that last thought}}}}</ref></ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">#</del>* <del class="diffchange diffchange-inline">This is actually an abuse of notation as </del>{{M|w<del class="diffchange diffchange-inline">^</del>{<del class="diffchange diffchange-inline">-</del>1}(x<del class="diffchange diffchange-inline">\in W</del>)}} is <del class="diffchange diffchange-inline">a subset </del>of {{M|<del class="diffchange diffchange-inline">X</del>}}<del class="diffchange diffchange-inline">, however it </del>is safe to use <del class="diffchange diffchange-inline">it </del>because <del class="diffchange diffchange-inline">(as is proved below) </del>{{M|f}} <del class="diffchange diffchange-inline">of any element of </del>{{M|w^{-1}(<del class="diffchange diffchange-inline">x\in W</del>)}} <del class="diffchange diffchange-inline">for a given </del>{{M|<del class="diffchange diffchange-inline">x</del>}} is <del class="diffchange diffchange-inline">the same. </del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">** We may also write </ins>{{M|<ins class="diffchange diffchange-inline">1=\tilde{f</ins>}<ins class="diffchange diffchange-inline">=f\circ w^{-1</ins>} <ins class="diffchange diffchange-inline">}} but this is a significant abuse of notation and should be avoided! It </ins>is safe to use <ins class="diffchange diffchange-inline">here </ins>because <ins class="diffchange diffchange-inline">of the "well-defined"-ness of </ins>{{M|<ins class="diffchange diffchange-inline">\tilde{</ins>f} }<ins class="diffchange diffchange-inline">}</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"># The function </del>{{M|\tilde{f} }} is unique <del class="diffchange diffchange-inline">if </del>{{M|<del class="diffchange diffchange-inline">w</del>}} is [[<del class="diffchange diffchange-inline">Surjection</del>|surjective]]</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">We may then say: </ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>==<del class="diffchange diffchange-inline">=Points to remember=</del>==</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">* "''</ins>{{M|<ins class="diffchange diffchange-inline">f}} may be factored through {{M|w}} to {{M|\tilde{f} }}''" or "{{M|f}} descends to the quotient via {{M|w}} to give {{M|\tilde{f} }}" </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">'''Claims: '''</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># {{M|\tilde{f}:W\rightarrow Y}} is given unambiguously by {{M|\tilde{f}:v\mapsto f(</ins>w^{-1}(<ins class="diffchange diffchange-inline">v)</ins>)}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># If </ins>{{M|<ins class="diffchange diffchange-inline">w:X\rightarrow W</ins>}} is <ins class="diffchange diffchange-inline">[[surjective]] then </ins>{{M|\tilde{f}<ins class="diffchange diffchange-inline">:W\rightarrow Y</ins>}} is unique <ins class="diffchange diffchange-inline">- the only function </ins>{{M|<ins class="diffchange diffchange-inline">(:W\rightarrow Y)}} such that the diagram commutes</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"># If {{M|f:X\rightarrow Y</ins>}} is [[<ins class="diffchange diffchange-inline">surjective]] then {{M</ins>|<ins class="diffchange diffchange-inline">\tilde{f}:W\rightarrow Y}} is [[</ins>surjective]] <ins class="diffchange diffchange-inline">also</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>==<ins class="diffchange diffchange-inline">Caveats</ins>==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">The following are good points to keep in mind when dealing with situations like this:</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Remembering the requirements:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* Remembering the requirements:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*: We want to induce a function {{M|\tilde{f}:W\rightarrow Y}} <del class="diffchange diffchange-inline">- </del>if {{M|1=w(x)=w(y)}} then {{M|1=\tilde{f}(w(x))=\tilde{f}(w(y))}} just by composition<del class="diffchange diffchange-inline">. </del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*: We want to induce a function {{M|\tilde{f}:W\rightarrow Y}} <ins class="diffchange diffchange-inline">such that all the information of {{M|f}} is "distilled" into {{M|w}}, notice that:</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*: <del class="diffchange diffchange-inline">If </del>{{M|1=f(x)\ne f(y)}} we're screwed <del class="diffchange diffchange-inline">in </del>this <del class="diffchange diffchange-inline">case</del>. So it is easy to see that we <del class="diffchange diffchange-inline">must have </del>{{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} <del class="diffchange diffchange-inline">otherwise we cannot </del>proceed.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*:* </ins>if {{M|1=w(x)=w(y)}} then {{M|1=\tilde{f}(w(x))=\tilde{f}(w(y))}} just by composition <ins class="diffchange diffchange-inline">of {{plural|function|s}}, regardless of {{M|\tilde{f} }}!</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*:<ins class="diffchange diffchange-inline">* so if </ins>{{M|1=f(x)\ne f(y)}} <ins class="diffchange diffchange-inline">but {{M|1=w(x)=w(y)}} then </ins>we're screwed <ins class="diffchange diffchange-inline">and cannot use </ins>this.  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*: </ins>So it is easy to see that we <ins class="diffchange diffchange-inline">require </ins>{{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} <ins class="diffchange diffchange-inline">in order to </ins>proceed.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Requires proof|grade=A|msg=Most of the proofs are done, I've done the surjective one like 3 times (CHECK THE TALK PAGE! SO YOU DON'T DO IT A FOURTH!) Also:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Move the proofs into sub-pages. It is just so much neater!}}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Begin Theorem}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Begin Theorem}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Claim: the induced function, {{M|\tilde{f} }} exists and is given unambiguously by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Claim: the induced function, {{M|\tilde{f} }} exists and is given unambiguously by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 73:</td>
<td colspan="2" class="diff-lineno">Line 83:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==See also==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Passing to the quotient]] - disambiguation page</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Equivalent conditions to being constant on the fibres of a map]]</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Todo|Factoring a map through the canonical projection of the equivalence relation it generates}}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">==Notes==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references group="Note"/></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{<del class="diffchange diffchange-inline">Definition</del>|<del class="diffchange diffchange-inline">Abstract Algebra</del>}}</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{<ins class="diffchange diffchange-inline">Theorem Of</ins>|<ins class="diffchange diffchange-inline">Elementary Set Theory|Set Theory</ins>}}</div></td></tr>
</table>
Alec
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=3210&oldid=prev
Alec: Alec moved page Factor (function) to Passing to the quotient (function): Name makes more sense
2016-10-08T17:42:27Z
<p>Alec moved page <a href="/index.php?title=Factor_(function)" class="mw-redirect" title="Factor (function)">Factor (function)</a> to <a href="/index.php?title=Passing_to_the_quotient_(function)" title="Passing to the quotient (function)">Passing to the quotient (function)</a>: Name makes more sense</p>
<table class='diff diff-contentalign-left'>
<tr style='vertical-align: top;'>
<td colspan='1' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='1' style="background-color: white; color:black; text-align: center;">Revision as of 17:42, 8 October 2016</td>
</tr><tr><td colspan='2' style='text-align: center;'><div class="mw-diff-empty">(No difference)</div>
</td></tr></table>
Alec
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=2699&oldid=prev
Alec: Tidied up a bit, fixed a typo or two.
2016-07-10T16:18:31Z
<p>Tidied up a bit, fixed a typo or two.</p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 16:18, 10 July 2016</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 57:</td>
<td colspan="2" class="diff-lineno">Line 57:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>'''Uniqueness'''</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>'''Uniqueness'''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>: Suppose another function exists, <math>\tilde{f}':W\rightarrow Y</math> that isn't the same as <math>\tilde{f}:W\rightarrow Y</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>: Suppose another function exists, <math>\tilde{f}':W\rightarrow Y</math> that isn't the same as <math>\tilde{f}:W\rightarrow Y</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:: That means <math>\exists u\in W:[\tilde{f}(u)\ne\tilde{f}'(u)]</math> <del class="diffchange diffchange-inline">(and </del>as {{M|w}} is <del class="diffchange diffchange-inline">''</del>surjective<del class="diffchange diffchange-inline">'' </del>{{M|1=\exists x\in X[<del class="diffchange diffchange-inline">p</del>(x)=u]}}<del class="diffchange diffchange-inline">)</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:: That means <math>\exists u\in W:[\tilde{f}(u)\ne\tilde{f}'(u)]</math></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:: <del class="diffchange diffchange-inline">Both </del>{{M|\tilde{f} }} and {{M|\tilde{f}'}} have the property of <math>f=\tilde{f}\circ w=\tilde{f}'\circ w</math> so:</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">::* Note, </ins>as {{M|w<ins class="diffchange diffchange-inline">:X\rightarrow W</ins>}} is <ins class="diffchange diffchange-inline">[[</ins>surjective<ins class="diffchange diffchange-inline">]], that </ins>{{M|1=\exists x<ins class="diffchange diffchange-inline">'</ins>\in X[<ins class="diffchange diffchange-inline">w</ins>(x<ins class="diffchange diffchange-inline">'</ins>)=u]}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>::: {{M|f(x)=\tilde{f}(<del class="diffchange diffchange-inline">p</del>(x))=\tilde{f}'(<del class="diffchange diffchange-inline">p</del>(x))}} <del class="diffchange diffchange-inline">by hypothesis</del>, <del class="diffchange diffchange-inline">for all </del>{{M|x}} <del class="diffchange diffchange-inline">however</del>, <del class="diffchange diffchange-inline">we know </del>{{M|\tilde{f} }} and {{M|\tilde{f}'}} <del class="diffchange diffchange-inline">don't agree over their entire domain, the </del>{{M|<del class="diffchange diffchange-inline">p</del>(x)}<del class="diffchange diffchange-inline">} they do not agree on violate this property </del>(as {{M|<del class="diffchange diffchange-inline">f</del>}} <del class="diffchange diffchange-inline">cannot be two things for </del>a <del class="diffchange diffchange-inline">given </del>{{M|<del class="diffchange diffchange-inline">x</del>}}<del class="diffchange diffchange-inline">)</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:: <ins class="diffchange diffchange-inline">However for both </ins>{{M|\tilde{f} }} and {{M|\tilde{f}'}} <ins class="diffchange diffchange-inline">we </ins>have the property of <math>f=\tilde{f}\circ w=\tilde{f}'\circ w</math> so:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">:: This contradicts that </del>{{M|\tilde{f} }} and {{M|\tilde{f}'}} <del class="diffchange diffchange-inline">are </del>different</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>::<ins class="diffchange diffchange-inline">: By hypothesis we have</ins>: {{M|<ins class="diffchange diffchange-inline">1=\forall x\in X[</ins>f(x)=\tilde{f}(<ins class="diffchange diffchange-inline">w</ins>(x))=\tilde{f}'(<ins class="diffchange diffchange-inline">w</ins>(x)<ins class="diffchange diffchange-inline">)]}} however we know:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:::* {{M|1=\exists x'\in X[w(x')=u]}} and {{M|1=\tilde{f}(u)\ne \tilde{f}'(u</ins>)}}, <ins class="diffchange diffchange-inline">this means:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:::** </ins>{{M|<ins class="diffchange diffchange-inline">1=f(</ins>x<ins class="diffchange diffchange-inline">')=\tilde{f</ins>}<ins class="diffchange diffchange-inline">(w(x'))\ne\tilde{f</ins>}<ins class="diffchange diffchange-inline">'(w(x'))}} - which contradicts the hypothesis.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:: However if {{M|w}} is not surjective</ins>, <ins class="diffchange diffchange-inline">then the parts of the domain on which </ins>{{M|\tilde{f} }} and {{M|\tilde{f}'}} <ins class="diffchange diffchange-inline">disagree on may never actually come up; that is to say:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">::* </ins>{{M|<ins class="diffchange diffchange-inline">1=\forall x\in X[\tilde{f}(w</ins>(x)<ins class="diffchange diffchange-inline">)=\tilde{f</ins>}<ins class="diffchange diffchange-inline">'</ins>(<ins class="diffchange diffchange-inline">w(x))]}} </ins>as <ins class="diffchange diffchange-inline">{{m|w:X\rightarrow W}} may never take an </ins>{{M|<ins class="diffchange diffchange-inline">x\in X</ins>}} <ins class="diffchange diffchange-inline">to </ins>a {{M|<ins class="diffchange diffchange-inline">z\in W</ins>}} <ins class="diffchange diffchange-inline">where </ins>{{M|\tilde{f}<ins class="diffchange diffchange-inline">(z)</ins>}} and {{M|\tilde{f}'<ins class="diffchange diffchange-inline">(z)</ins>}} <ins class="diffchange diffchange-inline">differ; ''but'' they could still be </ins>different <ins class="diffchange diffchange-inline">functions. </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
Alec
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=1349&oldid=prev
Alec: Made the diagram a bit neater, added in some points to make it easier to remember
2015-11-19T00:19:07Z
<p>Made the diagram a bit neater, added in some points to make it easier to remember</p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 00:19, 19 November 2015</td>
</tr><tr><td colspan="2" class="diff-lineno">Line 3:</td>
<td colspan="2" class="diff-lineno">Line 3:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>: '''{{M|f}} may be factored through {{M|w}}'''</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>: '''{{M|f}} may be factored through {{M|w}}'''</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>if {{M|f}} and {{M|w}} are such that:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>if {{M|f}} and {{M|w}} are such that:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* <math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math> (this is the same as: <math>\forall x,y\in X[f(x)\ne f(y)\implies w(x)\ne w(y)]</math>)</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* <math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math>  </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">*: </ins>(this is the same as: <math>\forall x,y\in X[f(x)\ne f(y)\implies w(x)\ne w(y)]</math>)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Then {{M|f}} ''induces'' a function, {{M|\tilde{f} }} such that <math>f=\tilde{f}\circ w</math>, or more simply that the following [[Commutative diagram|diagram commutes]]:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Then {{M|f}} ''induces'' a function, {{M|\tilde{f} }} such that <math>f=\tilde{f}\circ w</math>, or more simply that the following [[Commutative diagram|diagram commutes]]:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:<del class="diffchange diffchange-inline">: </del><math></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">{| class="wikitable" border="1"</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">| style="font-size</ins>:<ins class="diffchange diffchange-inline">1.5em;" | </ins><math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{xy}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\begin{xy}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\xymatrix{</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\xymatrix{</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 13:</td>
<td colspan="2" class="diff-lineno">Line 16:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{xy}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>\end{xy}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">! Diagram</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Note:</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>Note:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div># {{M|\tilde{f} }} may be explicitly written as {{M|\tilde{f}:W\rightarrow Y}} by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div># {{M|\tilde{f} }} may be explicitly written as {{M|\tilde{f}:W\rightarrow Y}} by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#* Or indeed {{M|1=\tilde{f}:=f\circ w^{-1} }}</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">#* This is actually an abuse of notation as {{M|w^{-1}(x\in W)}} is a subset of {{M|X}}, however it is safe to use it because (as is proved below) {{M|f}} of any element of {{M|w^{-1}(x\in W)}} for a given {{M|x}} is the same. </ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div># The function {{M|\tilde{f} }} is unique if {{M|w}} is [[Surjection|surjective]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div># The function {{M|\tilde{f} }} is unique if {{M|w}} is [[Surjection|surjective]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Points to remember===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Remembering the requirements:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*: We want to induce a function {{M|\tilde{f}:W\rightarrow Y}} - if {{M|1=w(x)=w(y)}} then {{M|1=\tilde{f}(w(x))=\tilde{f}(w(y))}} just by composition. </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*: If {{M|1=f(x)\ne f(y)}} we're screwed in this case. So it is easy to see that we must have {{M|1=[w(x)=w(y)]\implies[f(x)=f(y)]}} otherwise we cannot proceed.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==Proof of claims==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Begin Theorem}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Begin Theorem}}</div></td></tr>
<tr><td colspan="2" class="diff-lineno">Line 57:</td>
<td colspan="2" class="diff-lineno">Line 69:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Proof}}</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{End Theorem}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Definition|Abstract Algebra}}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>{{Definition|Abstract Algebra}}</div></td></tr>
</table>
Alec
http://www.maths.kisogo.com/index.php?title=Passing_to_the_quotient_(function)&diff=880&oldid=prev
Alec: Created page with "==Definition== Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}} (I have chosen {{M|W}} to mean "whatever") we can say: : '''{{M|f}} may be..."
2015-06-05T23:55:34Z
<p>Created page with "==Definition== Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}} (I have chosen {{M|W}} to mean "whatever") we can say: : '''{{M|f}} may be..."</p>
<p><b>New page</b></p><div>==Definition==<br />
Given a function, {{M|f:X\rightarrow Y}} and another function, {{M|w:X\rightarrow W}} (I have chosen {{M|W}} to mean "whatever") we can say:<br />
: '''{{M|f}} may be factored through {{M|w}}'''<br />
if {{M|f}} and {{M|w}} are such that:<br />
* <math>\forall x,y\in X[w(x)=w(y)\implies f(x)=f(y)]</math> (this is the same as: <math>\forall x,y\in X[f(x)\ne f(y)\implies w(x)\ne w(y)]</math>)<br />
Then {{M|f}} ''induces'' a function, {{M|\tilde{f} }} such that <math>f=\tilde{f}\circ w</math>, or more simply that the following [[Commutative diagram|diagram commutes]]:<br />
:: <math><br />
\begin{xy}<br />
\xymatrix{<br />
X \ar[r]^w \ar[dr]_f & W \ar@{.>}[d]^{\tilde{f}}\\<br />
& Y<br />
}<br />
\end{xy}<br />
</math><br />
Note:<br />
# {{M|\tilde{f} }} may be explicitly written as {{M|\tilde{f}:W\rightarrow Y}} by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}<br />
# The function {{M|\tilde{f} }} is unique if {{M|w}} is [[Surjection|surjective]]<br />
==Proof of claims==<br />
{{Begin Theorem}}<br />
Claim: the induced function, {{M|\tilde{f} }} exists and is given unambiguously by {{M|\tilde{f}:v\mapsto f(w^{-1}(v))}}<br />
{{Begin Proof}}<br />
'''Existence'''<br />
: Let {{M|\tilde{f}:W\rightarrow Y}} be given by: {{M|f:v\mapsto f(w^{-1}(v))}} - I need to prove this is a [[Function]]<br />
:: This means I must check it is well defined, a function must associate each point in its domain with exactly 1 element of its codomain<br />
::: Let {{M|v\in W}} be given<br />
:::: Let {{M|a\in w^{-1}(v)}} be given<br />
::::: Let {{M|b\in w^{-1}(v)}} be given<br />
:::::: We know <math>\forall a\in w^{-1}(v)</math> that <math>w(a)=v</math> by definition of <math>w^{-1}</math><br />
:::::: This means <math>w(a)=w(b)</math><br />
::::::: But by hypothesis <math>w(a)=w(b)\implies f(a)=f(b)</math><br />
:::::: So <math>f(a)=f(b)</math><br />
::::: Thus given an {{M|a\in w^{-1}(v)}}, <math>\forall b\in w^{-1}[f(a)=f(b)]</math><br />
:::: We now know (formally) that: (given a {{M|v}}) <math>\exists y\in Y\forall a\in w^{-1}(v)[f(a)=y]</math> - notice the <math>\exists y</math> comes first. We can uniquely define <math>f(w^{-1}(v))</math><br />
::: Since {{M|v\in W}} was arbitrary we know <math>\forall v\in W\exists y\in Y\forall a\in w^{-1}(v)[f(a)=y]</math><br />
:: We have now shown that <math>\tilde{f}</math> can be well defined (as the function that maps a {{M|v\in W}} to a {{M|y\in Y}}.<br />
:: To calculate <math>\tilde{f}(v)</math> we may choose any <math>a\in w^{-1}(v)</math> and define <math>\tilde{f}(v)=f(a)</math> - we know <math>f(a)</math> is the same for whichever <math>a\in w^{-1}(v)</math> we choose.<br />
: So we know the function <math>\tilde{f}:W\rightarrow Y</math> given by <math>\tilde{f}:x\mapsto f(w^{-1}(x))</math> exists<br />
<br />
<br />
This completes the proof<ref name="Alec">This is my (Alec's) own work</ref><br />
{{End Proof}}{{End Theorem}}<br />
{{Begin Theorem}}<br />
Claim: if {{M|w}} is surjective then the induced {{M|\tilde{f} }} is unique<br />
{{Begin Proof}}<br />
'''Uniqueness'''<br />
: Suppose another function exists, <math>\tilde{f}':W\rightarrow Y</math> that isn't the same as <math>\tilde{f}:W\rightarrow Y</math><br />
:: That means <math>\exists u\in W:[\tilde{f}(u)\ne\tilde{f}'(u)]</math> (and as {{M|w}} is ''surjective'' {{M|1=\exists x\in X[p(x)=u]}})<br />
:: Both {{M|\tilde{f} }} and {{M|\tilde{f}'}} have the property of <math>f=\tilde{f}\circ w=\tilde{f}'\circ w</math> so:<br />
::: {{M|f(x)=\tilde{f}(p(x))=\tilde{f}'(p(x))}} by hypothesis, for all {{M|x}} however, we know {{M|\tilde{f} }} and {{M|\tilde{f}'}} don't agree over their entire domain, the {{M|p(x)}} they do not agree on violate this property (as {{M|f}} cannot be two things for a given {{M|x}})<br />
:: This contradicts that {{M|\tilde{f} }} and {{M|\tilde{f}'}} are different<br />
<br />
<br />
This completes the proof<ref name="Alec"/><br />
<br />
: '''Notes:'''<br />
:# Notice that if {{M|w}} is not surjective, the point(s) on which {{M|\tilde{f} }} and {{M|\tilde{f}'}} disagree on may never actually come up, so it is indeed not-unique if {{M|w}} isn't surjective. <br />
{{End Proof}}<br />
{{End Theorem}}<br />
<br />
==References==<br />
<references/><br />
<br />
{{Definition|Abstract Algebra}}</div>
Alec