 Propuesto 1 |
|
|
|
|
|
|
  |
 Aug 19 2007, 09:31 PM
Publicado:
#1
|
|
 Dios Matemático Supremo  Grupo: Usuario FMAT Mensajes: 670 Registrado: 30-January 06 Desde: Ñuñoa, Santiago Miembro Nº: 524 Nacionalidad:  Universidad:  Sexo:   |
![TEX: \noindent Considere $A=\mathbb{Z}\times\mathbb{Z}$. Se define la relaci\'on $\Re$ en $A$ por:\\<br />$$(a_1,a_2) \ \Re \ (b_1,b_2) \iff \exists k \in \mathbb{Z}: a_1+a_2-b_1-b_2=2k$$\\<br />(i) Pruebe que $\Re$ es una relaci\'on de equivalencia.\\<br />(ii) Calcular expl\'icitamente $[(0,0)]_\Re$ y $[(1,0)]_\Re$.\\<br />(iii) Pruebe que $A=[(0,0)]_\Re \cup [(1,0)]_\Re$.\\<br />(iv) Pruebe que existe una biyecci\'on $f: [(1,0)]_\Re \to [(0,0)]_\Re$.](./tex/1187577153.gif)
--------------------    |
 |
 
|
|
Aug 19 2007, 11:03 PM
Publicado:
#2
|
|
 Dios Matemático Supremo Grupo: Usuario FMAT Mensajes: 1.112 Registrado: 21-December 05 Desde: El Bosque - Stgo Miembro Nº: 473 Colegio/Liceo:  Universidad:  Sexo: |
![TEX: \[<br />\begin{gathered}<br /> {\text{Voy a hacer el (i)}}{\text{. Y para lograrlo debo probar que }}\Re {\text{ es refleja}}{\text{, }} \hfill \\<br /> {\text{sim\'etrica y transitiva}}{\text{.}} \hfill \\<br /> \hfill \\<br /> \left. {\underline {\, <br /> {{\text{Reflexividad}}} \,}}\! \right| \hfill \\<br /> \hfill \\<br /> {\text{pd: }}(a_1 ,b_1 )\Re (a_1 ,b_1 ) \hfill \\<br /> \hfill \\<br /> (a_1 ,b_1 )\Re (a_1 ,b_1 ) \Leftrightarrow \exists k \in \mathbb{Z}:a_1 + b_1 - a_1 - b_1 = 2k \hfill \\<br /> \hfill \\<br /> {\text{Como }}\left( {a_1 + b_1 - a_1 - b_1 = 0 = 2 \cdot 0{\text{ }} \wedge {\text{ }}0 \in \mathbb{Z}} \right) \Rightarrow (a_1 ,b_1 )\Re (a_1 ,b_1 ) \Rightarrow \Re {\text{ es refleja}} \hfill \\ <br />\end{gathered} <br />\]<br />](./tex/1187582635.gif) ![TEX: \[<br />\begin{gathered}<br /> \left. {\underline {\, <br /> {{\text{Simetr\'ia}}} \,}}\! \right| \hfill \\<br /> \hfill \\<br /> {\text{Si }}(a_1 ,b_1 )\Re (a_2 ,b_2 ),{\text{ pd }}(a_2 ,b_2 )\Re (a_1 ,b_1 ) \hfill \\<br /> \hfill \\<br /> (a_1 ,b_1 )\Re (a_2 ,b_2 ) \Leftrightarrow \exists p \in \mathbb{Z}:a_1 + b_1 - a_2 - b_2 = 2p \Leftrightarrow \exists p \in \mathbb{Z}:a_1 + b_1 - (a_2 + b_2 ) = 2p \hfill \\<br /> \Leftrightarrow (a_2 + b_2 ) - (a_1 + b_1 ) = - 2p = 2 \cdot ( - p) \hfill \\<br /> \hfill \\<br /> {\text{Como }} - p \in \mathbb{Z} \Rightarrow {\text{ }}\exists k = - p \in \mathbb{Z}:a_2 + b_2 - a_1 - b_1 = 2k \Leftrightarrow (a_2 ,b_2 )\Re (a_1 ,b_1 ) \hfill \\<br /> \hfill \\<br /> {\text{Luego}}{\text{, es sim\'etrica}}{\text{.}} \hfill \\ <br />\end{gathered} <br />\]<br />](./tex/1187582637.gif) ![TEX: \[<br />\begin{gathered}<br /> \left. {\underline {\, <br /> {{\text{Transitividad}}} \,}}\! \right| \hfill \\<br /> \hfill \\<br /> {\text{Si }}(a_1 ,b_1 )\Re (a_2 ,b_2 ){\text{ y }}(a_2 ,b_2 )\Re (a_3 ,b_3 ){\text{, pd }}(a_1 ,b_1 )\Re (a_3 ,b_3 ) \hfill \\<br /> \hfill \\<br /> (a_1 ,b_1 )\Re (a_2 ,b_2 ) \Leftrightarrow \exists m \in \mathbb{Z}:a_1 + b_1 - a_2 - b_2 = 2m \hfill \\<br /> (a_2 ,b_2 )\Re (a_3 ,b_3 ) \Leftrightarrow \exists n \in \mathbb{Z}:a_2 + b_2 - a_3 - b_3 = 2n \hfill \\<br /> \hfill \\<br /> \Rightarrow (a_1 + b_1 - a_2 - b_2 ) + (a_2 + b_2 - a_3 - b_3 ) = 2m + 2n \hfill \\<br /> \Leftrightarrow a_1 + b_1 - a_3 - b_3 = 2(m + n) \hfill \\<br /> \hfill \\<br /> {\text{Como }}m + n \in \mathbb{Z} \Rightarrow \exists k = m + n \in \mathbb{Z}:a_1 + b_1 - a_3 - b_3 = 2(m + n) \Leftrightarrow (a_1 ,b_1 )\Re (a_3 ,b_3 ) \hfill \\<br /> \hfill \\<br /> {\text{Luego}}{\text{, }}\Re {\text{ es transitiva}} \hfill \\<br /> \hfill \\<br /> \therefore {\text{ }}\Re {\text{ es de equivalencia}} \hfill \\ <br />\end{gathered} <br />\]<br />](./tex/1187582640.gif) Saludos y buenas noches x_x -------------------- "El sentido común es el conjunto de todos los prejuicios adquiridos antes de los 18 años" A. Einstein.   Estudiante Ingeniería Civil Eléctrica - DIE USACH |
|
|
|
|
Jan 24 2010, 01:52 AM
Publicado:
#3
|
|
 Staff FMAT  Grupo: Super Moderador Mensajes: 4.857 Registrado: 2-January 08 Miembro Nº: 14.268 Nacionalidad: Colegio/Liceo:  Universidad: Sexo: |
![TEX: % MathType!MTEF!2!1!+-<br />% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn<br />% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr<br />% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9<br />% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x<br />% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGPb<br />% GaaeyAaiaabMcaaeaacaqGtbGaaeyzaiaabggacaqGGaGaaeiiamaa<br />% bmaabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyicI48aam<br />% WaaeaadaqadaqaaiaaicdacaGGSaGaaGimaaGaayjkaiaawMcaaaGa<br />% ay5waiaaw2faamaaBaaaleaacqGHCeIWaeqaaOGaeyO0H49aaeWaae<br />% aacaWGHbGaaiilaiaadkgaaiaawIcacaGLPaaacqGHCeIWdaqadaqa<br />% aiaaicdacaGGSaGaaGimaaGaayjkaiaawMcaaiabgkDiElabgoGiKi<br />% aadUgacqGHiiIZcqWIKeIOcaGG6aGaamyyaiabgUcaRiaadkgacqGH<br />% 9aqpcaaIYaGaam4AaiabgkDiElaadggacqGH9aqpcaaIYaGaam4Aai<br />% abgkHiTiaadkgaaeaacaqGmbGaaeyDaiaabwgacaqGNbGaae4Baiaa<br />% bccacaqG6aGaaeiiaiaabccadaWadaqaamaabmaabaGaaGimaiaacY<br />% cacaaIWaaacaGLOaGaayzkaaaacaGLBbGaayzxaaWaaSbaaSqaaiab<br />% gYricdqabaGccqGH9aqpdaGadaqaamaabmaabaGaamyyaiaacYcaca<br />% WGIbaacaGLOaGaayzkaaGaeyicI4SaeSijHi6aaWbaaSqabeaacaaI<br />% YaaaaOGaai4laiaadUgacqGHiiIZcqWIKeIOcaGG6aGaamyyaiabg2<br />% da9iaaikdacaWGRbGaeyOeI0IaamOyaaGaay5Eaiaaw2haaaqaaaqa<br />% aiaabofacaqGLbGaaeyyaiaabccadaqadaqaaiaadggacaGGSaGaam<br />% OyaaGaayjkaiaawMcaaiabgIGiopaadmaabaWaaeWaaeaacaaIXaGa<br />% aiilaiaaicdaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaWgaaWcba<br />% GaeyihHimabeaakiabgkDiEpaabmaabaGaamyyaiaacYcacaWGIbaa<br />% caGLOaGaayzkaaGaeyihHi8aaeWaaeaacaaIXaGaaiilaiaaicdaai<br />% aawIcacaGLPaaacqGHshI3cqGHdicjcaWGRbGaeyicI4SaeSijHiQa<br />% aiOoaiaadggacqGHRaWkcaWGIbGaeyOeI0IaaGymaiabg2da9iaaik<br />% dacaWGRbGaeyO0H4Taamyyaiabg2da9maabmaabaGaaGOmaiaadUga<br />% cqGHsislcaaIXaaacaGLOaGaayzkaaGaeyOeI0IaamOyaaqaaiaabY<br />% eacaqG1bGaaeyzaiaabEgacaqGVbGaaeiiaiaabQdacaqGGaGaaeii<br />% amaadmaabaWaaeWaaeaacaaIXaGaaiilaiaaicdaaiaawIcacaGLPa<br />% aaaiaawUfacaGLDbaadaWgaaWcbaGaeyihHimabeaakiabg2da9maa<br />% cmaabaWaaeWaaeaacaWGHbGaaiilaiaadkgaaiaawIcacaGLPaaacq<br />% GHiiIZcqWIKeIOdaahaaWcbeqaaiaaikdaaaGccaGGVaGaam4Aaiab<br />% gIGiolablssiIkaacQdacaWGHbGaeyypa0ZaaeWaaeaacaaIYaGaam<br />% 4AaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislcaWGIbaacaGL<br />% 7bGaayzFaaaaaaa!EBD8!<br />\[<br />\begin{gathered}<br /> {\text{ii)}} \hfill \\<br /> {\text{Sea }}\left( {a,b} \right) \in \left[ {\left( {0,0} \right)} \right]_\Re \Rightarrow \left( {a,b} \right)\Re \left( {0,0} \right) \Rightarrow \exists k \in \mathbb{Z}:a + b = 2k \Rightarrow a = 2k - b \hfill \\<br /> {\text{Luego : }}\left[ {\left( {0,0} \right)} \right]_\Re = \left\{ {\left( {a,b} \right) \in \mathbb{Z}^2 /k \in \mathbb{Z}:a = 2k - b} \right\} \hfill \\<br /> \hfill \\<br /> {\text{Sea }}\left( {a,b} \right) \in \left[ {\left( {1,0} \right)} \right]_\Re \Rightarrow \left( {a,b} \right)\Re \left( {1,0} \right) \Rightarrow \exists k \in \mathbb{Z}:a + b - 1 = 2k \Rightarrow a = \left( {2k - 1} \right) - b \hfill \\<br /> {\text{Luego : }}\left[ {\left( {1,0} \right)} \right]_\Re = \left\{ {\left( {a,b} \right) \in \mathbb{Z}^2 /k \in \mathbb{Z}:a = \left( {2k - 1} \right) - b} \right\} \hfill \\ <br />\end{gathered} <br />\]](./tex/b5a34a16d1a57225c41493e2eaa56273.png) ![TEX: % MathType!MTEF!2!1!+-<br />% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn<br />% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr<br />% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9<br />% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x<br />% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGPb<br />% GaaeyAaiaabMgacaqGPaaabaGaae4uaiaabwgacaqGHbGaaeiiaiaa<br />% bccadaqadaqaaiaadggacaGGSaGaamOyaaGaayjkaiaawMcaaiabgI<br />% GiopaadmaabaWaaeWaaeaacaaIXaGaaiilaiaaicdaaiaawIcacaGL<br />% PaaaaiaawUfacaGLDbaadaWgaaWcbaGaeyihHimabeaakiabgQIiip<br />% aadmaabaWaaeWaaeaacaaIWaGaaiilaiaaicdaaiaawIcacaGLPaaa<br />% aiaawUfacaGLDbaadaWgaaWcbaGaeyihHimabeaakiabgsDiBpaabm<br />% aabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyicI48aamWa<br />% aeaadaqadaqaaiaaigdacaGGSaGaaGimaaGaayjkaiaawMcaaaGaay<br />% 5waiaaw2faamaaBaaaleaacqGHCeIWaeqaaOGaeyikIO9aaeWaaeaa<br />% caWGHbGaaiilaiaadkgaaiaawIcacaGLPaaacqGHiiIZdaWadaqaam<br />% aabmaabaGaaGimaiaacYcacaaIWaaacaGLOaGaayzkaaaacaGLBbGa<br />% ayzxaaWaaSbaaSqaaiabgYricdqabaaakeaacqGHuhY2caWGRbGaey<br />% icI4SaeSijHiQaaiOoaiaadggacqGH9aqpcaaIYaGaam4AaiabgkHi<br />% TiaadkgacqGHOiI2caWGHbGaeyypa0ZaaeWaaeaacaaIYaGaam4Aai<br />% abgkHiTiaaigdaaiaawIcacaGLPaaacqGHsislcaWGIbaabaGaeyi1<br />% HSTaam4AaiabgIGiolablssiIkaacQdacaWGHbGaey4kaSIaamOyai<br />% abg2da9iaaikdacaWGRbGaeyikIOTaamyyaiabgUcaRiaadkgacqGH<br />% 9aqpcaaIYaGaam4AaiabgkHiTiaaigdaaaaa!9CF9!<br />\[<br />\begin{gathered}<br /> {\text{iii)}} \hfill \\<br /> {\text{Sea }}\left( {a,b} \right) \in \left[ {\left( {1,0} \right)} \right]_\Re \cup \left[ {\left( {0,0} \right)} \right]_\Re \Leftrightarrow \left( {a,b} \right) \in \left[ {\left( {1,0} \right)} \right]_\Re \vee \left( {a,b} \right) \in \left[ {\left( {0,0} \right)} \right]_\Re \hfill \\<br /> \Leftrightarrow k \in \mathbb{Z}:a = 2k - b \vee a = \left( {2k - 1} \right) - b \hfill \\<br /> \Leftrightarrow k \in \mathbb{Z}:a + b = 2k \vee a + b = 2k - 1 \hfill \\ <br />\end{gathered} <br />\]](./tex/e119b45ca3046adf4bbd2d70e359ba47.png) ![TEX: % MathType!MTEF!2!1!+-<br />% feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn<br />% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr<br />% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9<br />% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x<br />% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGmb<br />% GaaeyDaiaabwgacaqGNbGaae4BaiaabQdacaqGGaWaamWaaeaadaqa<br />% daqaaiaaigdacaGGSaGaaGimaaGaayjkaiaawMcaaaGaay5waiaaw2<br />% faamaaBaaaleaacqGHCeIWaeqaaOGaeyOkIG8aamWaaeaadaqadaqa<br />% aiaaicdacaGGSaGaaGimaaGaayjkaiaawMcaaaGaay5waiaaw2faam<br />% aaBaaaleaacqGHCeIWaeqaaOGaeyypa0ZaaiWaaeaadaqadaqaaiaa<br />% dggacaGGSaGaamOyaaGaayjkaiaawMcaaiabgIGiolablssiIoaaCa<br />% aaleqabaGaaGOmaaaakiaac+cacaWGRbGaeyicI4SaeSijHiQaaiOo<br />% aiaadggacqGHRaWkcaWGIbGaeyypa0JaaGOmaiaadUgacqGHOiI2ca<br />% WGHbGaey4kaSIaamOyaiabg2da9iaaikdacaWGRbGaeyOeI0IaaGym<br />% aaGaay5Eaiaaw2haaaqaaiaabgeacaqGZbGaaeyAaiaabccacaqGZb<br />% GaaeyzaiaabccacaqGZbGaaeyAaiaabEgacaqG1bGaaeyzaiaabcca<br />% caqGXbGaaeyDaiaabwgacaqGGaGaaeiiamaadmaabaWaaeWaaeaaca<br />% aIXaGaaiilaiaaicdaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaWg<br />% aaWcbaGaeyihHimabeaakiabgQIiipaadmaabaWaaeWaaeaacaaIWa<br />% GaaiilaiaaicdaaiaawIcacaGLPaaaaiaawUfacaGLDbaadaWgaaWc<br />% baGaeyihHimabeaakiabg2da9iablssiIkaadIhacqWIKeIOaeaaae<br />% aacaqGPbGaaeODaiaabMcaaeaacaqGtbGaaeyzaiaabggacaqGGaGa<br />% aeiiaiaadAgacaGG6aWaamWaaeaadaqadaqaaiaaigdacaGGSaGaaG<br />% imaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaBaaaleaacqGHCeIW<br />% aeqaaOGaeyOKH46aamWaaeaadaqadaqaaiaaicdacaGGSaGaaGimaa<br />% GaayjkaiaawMcaaaGaay5waiaaw2faamaaBaaaleaacqGHCeIWaeqa<br />% aOGaaeilaiaabccacaqGKbGaaeyzaiaabAgacaqGPbGaaeOBaiaabg<br />% gacaqGTbGaae4BaiaabohacaqGSbGaaeyyaiaabccacaqGJbGaae4B<br />% aiaab2gacaqGVbGaaeiiaiaabohacaqGPbGaae4zaiaabwhacaqGLb<br />% GaaeiiaiaabQdacaqGGaGaaeiiaiaadAgadaqadaqaaiaaikdacaWG<br />% RbGaeyOeI0IaaGymaiabgkHiTiaadkgacaGGSaGaamOyaaGaayjkai<br />% aawMcaaiabg2da9maabmaabaGaaGOmaiaadUgacqGHsislcaWGIbGa<br />% aiilaiaadkgaaiaawIcacaGLPaaaaeaacaqGbbGaae4CaiaabMgaca<br />% qGGaGaaeOoaaqaaiaadAgadaqadaqaaiaaikdacaWGRbWaaSbaaSqa<br />% aiaaigdaaeqaaOGaeyOeI0IaaGymaiabgkHiTiaadkgadaWgaaWcba<br />% GaaGymaaqabaGccaGGSaGaamOyamaaBaaaleaacaaIXaaabeaaaOGa<br />% ayjkaiaawMcaaiabg2da9iaadAgadaqadaqaaiaaikdacaWGRbWaaS<br />% baaSqaaiaaikdaaeqaaOGaeyOeI0IaaGymaiabgkHiTiaadkgadaWg<br />% aaWcbaGaaGOmaaqabaGccaGGSaGaamOyamaaBaaaleaacaaIYaaabe<br />% aaaOGaayjkaiaawMcaaiabgsDiBpaabmaabaGaaGOmaiaadUgadaWg<br />% aaWcbaGaaGymaaqabaGccqGHsislcaWGIbWaaSbaaSqaaiaaigdaae<br />% qaaOGaaiilaiaadkgadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGL<br />% PaaacqGH9aqpdaqadaqaaiaaikdacaWGRbWaaSbaaSqaaiaaikdaae<br />% qaaOGaeyOeI0IaamOyamaaBaaaleaacaaIYaaabeaakiaacYcacaWG<br />% IbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaGaamOzam<br />% aabmaabaGaaGOmaiaadUgadaWgaaWcbaGaaGymaaqabaGccqGHsisl<br />% caaIXaGaeyOeI0IaamOyamaaBaaaleaacaaIXaaabeaakiaacYcaca<br />% WGIbWaaSbaaSqaaiaaigdaaeqaaaGccaGLOaGaayzkaaGaeyypa0Ja<br />% amOzamaabmaabaGaaGOmaiaadUgadaWgaaWcbaGaaGOmaaqabaGccq<br />% GHsislcaaIXaGaeyOeI0IaamOyamaaBaaaleaacaaIYaaabeaakiaa<br />% cYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaey<br />% i1HS9aaucfaqaabeqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccqGH<br />% 9aqpcaWGIbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaGOmaiaadUgada<br />% WgaaWcbaGaaGymaaqabaGccqGHsislcaWGIbWaaSbaaSqaaiaaigda<br />% aeqaaOGaeyypa0JaaGOmaiaadUgadaWgaaWcbaGaaGOmaaqabaGccq<br />% GHsislcaWGIbWaaSbaaSqaaiaaikdaaeqaaaaaaaGcbaWaaeWaaeaa<br />% caaIYaGaam4AamaaBaaaleaacaaIXaaabeaakiabgkHiTiaaigdacq<br />% GHsislcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkgadaWg<br />% aaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGMbWaae<br />% WaaeaacaaIYaGaam4AamaaBaaaleaacaaIYaaabeaakiabgkHiTiaa<br />% igdacqGHsislcaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadk<br />% gadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacqGHuhY2caWG<br />% RbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Jaam4AamaaBaaaleaaca<br />% aIYaaabeaakiabgEIizlaadkgadaWgaaWcbaGaaGymaaqabaGccqGH<br />% 9aqpcaWGIbWaaSbaaSqaaiaaikdaaeqaaaGcbaWaaeWaaeaacaaIYa<br />% Gaam4AamaaBaaaleaacaaIXaaabeaakiabgkHiTiaaigdacqGHsisl<br />% caWGIbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadkgadaWgaaWcba<br />% GaaGymaaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGMbWaaeWaaeaa<br />% caaIYaGaam4AamaaBaaaleaacaaIYaaabeaakiabgkHiTiaaigdacq<br />% GHsislcaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadkgadaWg<br />% aaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaadaqadaqaaiaaikdaca<br />% WGRbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaaGymaiabgkHiTiaa<br />% dkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamOyamaaBaaaleaaca<br />% aIXaaabeaaaOGaayjkaiaawMcaamaabmaabaGaaGOmaiaadUgadaWg<br />% aaWcbaGaaGOmaaqabaGccqGHsislcaaIXaGaeyOeI0IaamOyamaaBa<br />% aaleaacaaIYaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqa<br />% aaGccaGLOaGaayzkaaaabaaabaGaaeywaiaabccacaqG0bGaaeyyai<br />% aab2gacaqGIbGaaeyAaiaabwgacaqGUbGaaeiiaiaabQdaaeaacaqG<br />% tbGaaeyzaiaabggacaqGGaGaaeiiamaabmaabaGaaGOmaiaadUgacq<br />% GHsislcaaIXaGaeyOeI0IaamOyaiaacYcacaWGIbaacaGLOaGaayzk<br />% aaGaeyicI4SaeSijHi6aaWbaaSqabeaacaaIYaaaaOGaaeilaiaabc<br />% cacaqGSbGaaeyDaiaabwgacaqGNbGaae4BaiaabccacaqG6aGaaeii<br />% aiaabccacaWGMbWaaeWaaeaacaaIYaGaam4AaiabgkHiTiaaigdacq<br />% GHsislcaWGIbGaaiilaiaadkgaaiaawIcacaGLPaaacqGH9aqpdaqa<br />% daqaaiaaikdacaWGRbGaeyOeI0IaamOyaiaacYcacaWGIbaacaGLOa<br />% GaayzkaaGaaeilaiaabccacaqGVbGaae4CaiaabwgacaqGHbGaaeii<br />% aiaabohacaqGPbGaaeyzaiaab2gacaqGWbGaaeOCaiaabwgacaqGGa<br />% GaaeyzaiaabohacaqGGaGaaeiCaiaab+gacaqGZbGaaeyAaiaabkga<br />% caqGSbGaaeyzaaqaaiaabwgacaqGUbGaae4yaiaab+gacaqGUbGaae<br />% iDaiaabkhacaqGHbGaaeiiaiaabwhacaqGUbGaaeiiaiaabwgacaqG<br />% SbGaaeyzaiaab2gacaqGLbGaaeOBaiaabshacaqGVbGaaeiiaiaabs<br />% gacaqGLbGaaeiiaiablssiIoaaCaaaleqabaGaaGOmaaaakiaabcca<br />% caqGWbGaaeyyaiaabkhacaqGHbGaaeiiaiaabogacaqG1bGaaeyyai<br />% aabYgacaqGXbGaaeyDaiaabMgacaqGLbGaaeOCaiaabccacaqGLbGa<br />% aeiBaiaabwgacaqGTbGaaeyzaiaab6gacaqG0bGaae4Baiaabccaca<br />% qGKbGaaeyzaiaabccacaqGSbGaaeiBaiaabwgacaqGNbGaaeyyaiaa<br />% bsgacaqGHbGaaeOlaaqaaaqaaiaabcfacaqGVbGaaeOCaiaabccaca<br />% qGSbGaae4BaiaabccacaqGHbGaaeOBaiaabshacaqGLbGaaeOCaiaa<br />% bMgacaqGVbGaaeOCaiaabYcacaqGGaGaaeyzaiaabIhacaqGPbGaae<br />% 4CaiaabshacaqGLbGaaeiiaiaabwhacaqGUbGaaeyyaiaabccacaqG<br />% IbGaaeyAaiaabMhacaqGLbGaae4yaiaabogacaqGPbGaae4Baiaab6<br />% gacaqGGaGaaeiiaiaadAgacaGG6aWaamWaaeaadaqadaqaaiaaigda<br />% caGGSaGaaGimaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaBaaale<br />% aacqGHCeIWaeqaaOGaeyOKH46aamWaaeaadaqadaqaaiaaicdacaGG<br />% SaGaaGimaaGaayjkaiaawMcaaaGaay5waiaaw2faamaaBaaaleaacq<br />% GHCeIWaeqaaaaaaa!38D7!<br />\[<br />\begin{gathered}<br /> {\text{Luego: }}\left[ {\left( {1,0} \right)} \right]_\Re \cup \left[ {\left( {0,0} \right)} \right]_\Re = \left\{ {\left( {a,b} \right) \in \mathbb{Z}^2 /k \in \mathbb{Z}:a + b = 2k \vee a + b = 2k - 1} \right\} \hfill \\<br /> {\text{Asi se sigue que }}\left[ {\left( {1,0} \right)} \right]_\Re \cup \left[ {\left( {0,0} \right)} \right]_\Re = \mathbb{Z}x\mathbb{Z} \hfill \\<br /> \hfill \\<br /> {\text{iv)}} \hfill \\<br /> {\text{Sea }}f:\left[ {\left( {1,0} \right)} \right]_\Re \to \left[ {\left( {0,0} \right)} \right]_\Re {\text{, definamosla como sigue : }}f\left( {2k - 1 - b,b} \right) = \left( {2k - b,b} \right) \hfill \\<br /> {\text{Asi :}} \hfill \\<br /> f\left( {2k_1 - 1 - b_1 ,b_1 } \right) = f\left( {2k_2 - 1 - b_2 ,b_2 } \right) \Leftrightarrow \left( {2k_1 - b_1 ,b_1 } \right) = \left( {2k_2 - b_2 ,b_2 } \right) \hfill \\<br /> f\left( {2k_1 - 1 - b_1 ,b_1 } \right) = f\left( {2k_2 - 1 - b_2 ,b_2 } \right) \Leftrightarrow \left. {\underline {\, <br /> \begin{gathered}<br /> b_1 = b_2 \hfill \\<br /> 2k_1 - b_1 = 2k_2 - b_2 \hfill \\ <br />\end{gathered} \,}}\! \right| \hfill \\<br /> \left( {2k_1 - 1 - b_1 ,b_1 } \right) = f\left( {2k_2 - 1 - b_2 ,b_2 } \right) \Leftrightarrow k_1 = k_2 \wedge b_1 = b_2 \hfill \\<br /> \left( {2k_1 - 1 - b_1 ,b_1 } \right) = f\left( {2k_2 - 1 - b_2 ,b_2 } \right)\left( {2k_1 - 1 - b_1 ,b_1 } \right)\left( {2k_2 - 1 - b_2 ,b_2 } \right) \hfill \\<br /> \hfill \\<br /> {\text{Y tambien :}} \hfill \\<br /> {\text{Sea }}\left( {2k - 1 - b,b} \right) \in \mathbb{Z}^2 {\text{, luego : }}f\left( {2k - 1 - b,b} \right) = \left( {2k - b,b} \right){\text{, osea siempre es posible}} \hfill \\<br /> {\text{encontra un elemento de }}\mathbb{Z}^2 {\text{ para cualquier elemento de llegada}}{\text{.}} \hfill \\<br /> \hfill \\<br /> {\text{Por lo anterior}}{\text{, existe una biyeccion }}f:\left[ {\left( {1,0} \right)} \right]_\Re \to \left[ {\left( {0,0} \right)} \right]_\Re \hfill \\ <br />\end{gathered} <br />\]](./tex/50fd25e38e009762e1ec40ba2fd6da47.png) -------------------- Candidato a doctor en Cs. De la ingeniería mención modelamiento matemático, DIM. Universidad de Chile
Magíster en ciencias mención matemática, Profesor de estado en matemáticas y computación, Licenciado en educación matemáticas y computación, USACH |
|
|
|
|
1 usuario(s) está(n) leyendo esta discusión (1 invitado(s) y 0 usuario(s) anónimo(s))
0 miembro(s):
| Versión Lo-Fi | Fecha y Hora actual: 3rd April 2025 - 09:58 PM |
