[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

Re: [obm-l] RE: [obm-l] Questão de Análise

   Oi Artur,
>
>Oi Duda!
>Se X_1,... e X_n estao em P(A), entao cada X_i esta contido s em Uniao =
>X_i.
>Pelas condicoes dadas, segue-se que F(Uniao X_i) estah contido em cada =
>um
>dos F(X_i). Logo, F(Uniao X_i) estah contido em Interseccao F(X_i). Alem
>disto, temos que Interseccao F(X_i) esta contido em cada um dos F(x_i), =
>o
>que acarreta que cada F(F(X_i)) =3D X_i esteja contido em F(Interseccao
>F(X_i)). Prosseguindo, temos que Uniao X_i esta contido em F(Interseccao
>F(X_i), o que implica que F(F(Interseccao F(X_i)) =3D Interseccao F(X_i)
>esteja contido em F(Uniao X_i), Assim  concluimos que F(Uni=E3o
>X_i) =3D Interse=E7=E3o F(X_i) --Ufa! .Interessante observar que isto eh =
>valido
>mesmo para subcolecoes nao numeraveis de P(A).=20
>
>Agora, temos que Interseccao X_i estah contido em cada X_i, de modo que =
>cada
>F(X_i) estah contido em F(Interseccao X_i). Logo, Uniao F(X_i) esta =
>contido
>em F(Interseccao X_i). Alem disto, cada F(X_i) estah contido  em Uniao
>F(x_i), de modo que F(Uniao F(X_i)) esta contido em cada um dos =
>F(F(X_i)) =3D
>X_i. Segue-se que F(Uniao F(X_i)) esta contido em Interseccao (X_i), do =
>que
>concluimos que F(Interseccao X_i) esta contido em F(F(Uniao F(X_i))) =3D =
>Uniao
>F(X_i). E assim, segue-se que F(Interse=E7=E3o X_i) =3D Uni=E3o F(X_i), =
>completando
>a prova. Verificamos de novo que isto eh valido mesmo para subcolecoes =
>nao
>numeraveis de P(A).
>
>Das condicoes dadas segue-se que F eh bijetora. Sendo 0 o conjunto =
>vazio,
>temos para todo X de P(A) que 0 estah contido em X e que, portanto, F(X)
>esta contido em F(0). Mas como F eh bijetora, para algum X temos F(X) =
>=3D A,
>de modo que F(0) =3D A. Logo, F(A) =3D F(F(0)) =3D 0. Isto nao prova, =
>mas
>desconfio que F eh a funcao complemento.

  Acho que nao e' sempre assim nao: seja f uma involucao de A e F(X)={f(x),
x no complementar de A}. Entao F satisfaz as condicoes do enunciado.
  Abracos,
           Gugu


>
>Um abraco!
>Artur=20
>
>> Ol=E1 Pessoal!
>>=20
>> Estou resolvendo o livro do Elon de An=E1lise e h=E1 um exerc=EDcio =
>que n=E3o
>> estou
>> conseguindo resolver.
>>=20
>> Seja A um conjunto e P(A) o conjunto das partes de A. Considere uma =
>fun=E7=E3o
>> f:P(A)->P(A) que satisfaz as propriedades: se X est=E1 contido em Y =
>(ambos
>> de
>> P(A)) ent=E3o F(Y) est=E1 contido em F(X); e F(F(X)) =3D X. Mostrar =
>que F(Uni=E3o
>> X_i) =3D Interse=E7=E3o F(X_i) e tamb=E9m F(Interse=E7=E3o X_i) =3D =
>Uni=E3o F(X_i).
>>=20
>> Uma fun=E7=E3o que satisfaz essas condi=E7=F5es =E9 F(X) =3D =
>Complementar X.
>>=20
>
>
>------=_NextPart_000_0006_01C37BF4.472A5860
>Content-Type: application/ms-tnef;
>	name="winmail.dat"
>Content-Transfer-Encoding: base64
>Content-Disposition: attachment;
>	filename="winmail.dat"
>
>eJ8+IjsEAQaQCAAEAAAAAAABAAEAAQeQBgAIAAAA5AQAAAAAAADoAAEIgAcAGAAAAElQTS5NaWNy
>b3NvZnQgTWFpbC5Ob3RlADEIAQ2ABAACAAAAAgACAAEGAAcAAQAAAAAAAAEGgAMADgAAANMHCQAQ
>AAEALQAAAAIAIwEBA5AGAMQJAAAuAAAACwACAAEAAAALACMAAAAAAAMAJgAAAAAACwApAAAAAAAD
>AC4AAAAAAAIBMQABAAAAGAAAAAAAAACIHK2dGmnOQL2OqXUAnoiTJDwmAAMANgAAAAAAHgBwAAEA
>AAAbAAAAW29ibS1sXSBRdWVzdONvIGRlIEFu4Wxpc2UAAAIBcQABAAAAFgAAAAHDfA0wO+z0EVOo
>XU0Vus68Qn6SA7cAAAIBHQwBAAAAGQAAAFNNVFA6QVJUVVJAT1BFTkRGLkNPTS5CUgAAAAALAAEO
>AAAAAEAABg4AriNJDXzDAQIBCg4BAAAAGAAAAAAAAACIHK2dGmnOQL2OqXUAnoiTwoAAAAMAFA4A
>AAAACwAfDgEAAAAeACgOAQAAADAAAAAwMDAwMDAwOAFhcnR1ckBvcGVuZGYuY29tLmJyAXBvcDMu
>b3BlbmRmLmNvbS5icgAeACkOAQAAADAAAAAwMDAwMDAwOAFhcnR1ckBvcGVuZGYuY29tLmJyAXBv
>cDMub3BlbmRmLmNvbS5icgACAQkQAQAAALYEAACyBAAAaAoAAExaRnVQQAcuAwAKAHJjcGcxMjXi
>MgNDdGV4BUEBAwH3TwqAAqQD4wIAY2gKwHPwZXQwIAcTAoAP8wBQfwRWCFUHshHFDlEDARDHMvcG
>AAbDEcUzBEYQyRLbEdPbCO8J9zsYvw4wNRHCDGDOYwBQCwkBZDM2EVALpgAgT2kgRHVkYQ4hCqIK
>gAZgIFhfMfQsLh7gIB6CA6AHkAGQAm8fEG0gUChBKc4sHxACMB+xY2Ed4B6Rzx2gH4IgwAIhaWQf
>wAQgPR/hVQMAH7EhIR8AUGXXC2AEICGxZA3gbweRHeCtHeBzIFARIGcKUC0RIEQgcQpQIEYoIocp
>3R9zaCGnH+Eg03Uf8CIADwQgJXAl8h8ATG9nb3cgUCV/JotJAjAEkBEgY+cg0B/AKBZBbB/hI8Af
>kP8owQ6wBGAEICUyKt8ppCaf/SeneCmBIFAfwCUyANAKwH8YwCGBJTIg0yjwKBQpoD35IRZlai7b
>KPAtfzLQIxH/A2AEECSxC4AiACy6Iocun/80jzCZB3ALUA3gMbQyYjk/vzLTO98pozOfKRggUEEE
>EPUHcCAhomMKQAdwLQUo8+BcJ2Uzbx4UKXM9Fw1CETdCEy4mLS1VZtkd8CAuPTMHkHMAcA6w5TDQ
>YhEgcnYKwTqTLIF/HxAmgEZQOwAiAQeBBGAg4wqxIQBzdWIYMQWRB5Eabh+xbiegBJBhdmV/BAAk
>IB6AIBIfAB4UHhRB/yiwSEAsvyslN9YmjzfCIFD/SiEEYTDkMhUpfzjPKVQodv8ihC4/UW9Sdywa
>T/8hpyJI/zBmTzoo9jKGLq8nazKLHwD/BmAkz1sfKk8rUjoFT4ZA7n9VXmBPMmJfezLiU2ofAEX/
>MTBAciSPQz9CxUHmOecFoP864TGRNoExMEgQA2BGUB8A3lYGcjsRTAJKIW5tUDp130bvR/9JD0oX
>SrpEI28EIOMkrG9iYmlqETBLkV5iz2zSEVAgsQIganUCMB/AeUZQemksp3BzLJAiAVh/ShYlIxFQ
>UL95UCUTIFBwtxhhRbEow1hk/yjhMChh/k1zw3BBdct70XCCB0BeoO97QkvkfJMzAEFZ3n4AgTK/
>KHh5wTMAMmJ+ADLiMB8A/klvMnGCbTMgUADASgIE8N8CIG3gWlVvYiEAZneQK1LvbFQHgHehcutV
>H/ABoHCQtwWgHgUHEHQIcEqrPh2A2xigQiAxIyEEEG8HQB4Fy4switZFLIF1IEVxBvBvSdBs0h/A
>OwB2A2BickX3GFADoEohQQuQi4E7ACUB/R6AaItzJ6EOwASQAOBCIPxkY4ZFj5JCRoswH4EIYO+K
>1iGxNjaNlnJy5YyoBmD9M6FBJ5J3V3mFdzl0sgqx/w6wSgOCwQhQAIEEgR6AJ6AnhwNDp4rWZjog
>Ei0+13mnRaAh4HNE4HpoUW0i720wCJAkMQeQOiSRe1IfkPuLc1wJWWIAbhAG4BCwiuW/AQCK1iAS
>UJECMEPmWVCT+56ufJI7HxEyYzLUfiEn4P50cJBGdEHOizBCz0PcHoD3AZAG0EIROTj5aj9rTZTv
>P4kQmhycPEWCc9WnxGY1/weRqTKA5ghQh7YKwaTQjEwFHhR9s3AAAB4AQhABAAAAKAAAADwwMDBk
>MDFjMzdiZjAkYWQ3YjkxZjAkMDMwMWE4YzBAc3RhYmVsPgADAJIQAQAAAAIBFDoBAAAAEAAAAImu
>tjCdtH1NlIWMEW+Y5acDAN4/r28AAAMACVkBAAAAAwBAZQAAAAALABOACCAGAAAAAADAAAAAAAAA
>RgAAAAADhQAAAAAAAAMAFYAIIAYAAAAAAMAAAAAAAABGAAAAABCFAAAAAAAAAwAbgAggBgAAAAAA
>wAAAAAAAAEYAAAAAUoUAAOOQAQADAEmACCAGAAAAAADAAAAAAAAARgAAAAABhQAAAAAAAEAASoAI
>IAYAAAAAAMAAAAAAAABGAAAAAGCFAAAAOE0lGQAAAB4Ad4AIIAYAAAAAAMAAAAAAAABGAAAAAFSF
>AAABAAAABQAAADEwLjAAAAAACwB4gAggBgAAAAAAwAAAAAAAAEYAAAAABoUAAAAAAAALAHyACCAG
>AAAAAADAAAAAAAAARgAAAAAOhQAAAAAAAAMAf4AIIAYAAAAAAMAAAAAAAABGAAAAABiFAAAAAAAA
>CwCUgAggBgAAAAAAwAAAAAAAAEYAAAAAgoUAAAEAAAACAfgPAQAAABAAAACIHK2dGmnOQL2OqXUA
>noiTAgH6DwEAAAAQAAAAiBytnRppzkC9jql1AJ6IkwIB+w8BAAAAcAAAAAAAAAA4obsQBeUQGqG7
>CAArKlbCAABtc3BzdC5kbGwAAAAAAE5JVEH5v7gBAKoAN9luAAAAQzpcV0lORE9XU1xBcHBsaWNh
>dGlvbiBEYXRhXE1pY3Jvc29mdFxPdXRsb29rXE91dGxvb2sxLnBzdAADAP4PBQAAAAMADTT9NwIA
>AgEUNAEAAAAQAAAATklUQfm/uAEAqgA32W4AAAIBfwABAAAAMQAAADAwMDAwMDAwODgxQ0FEOUQx
>QTY5Q0U0MEJEOEVBOTc1MDA5RTg4OTM2NDNDMjYwMAAAAAADAAYQip34EwMABxCoBgAAAwAQEAAA
>AAADABEQAgAAAB4ACBABAAAAZQAAAE9JRFVEQVNFWDEsRVhORVNUQU9FTVAoQSksRU5UQU9DQURB
>WElFU1RBQ09OVElET1NFTVVOSUFPWElQRUxBU0NPTkRJQ09FU0RBREFTLFNFR1VFLVNFUVVFRihV
>TklBT1hJKUUAAAAAJp0=
>
>------=_NextPart_000_0006_01C37BF4.472A5860--
>
>=========================================================================
>Instrugues para entrar na lista, sair da lista e usar a lista em
>http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html
>=========================================================================

=========================================================================
Instruções para entrar na lista, sair da lista e usar a lista em
http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html
=========================================================================