❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠
▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧
❯♠❛ Pr♦♣♦st❛ ❞❡ ❖✜❝✐♥❛ s♦❜r❡ ❈ó❞✐❣♦s ♣❛r❛ ❛
❈♦♥t❡①t✉❛❧✐③❛çã♦ ❞♦ ❊st✉❞♦ ❞❡ ❆r✐t♠ét✐❝❛ ❡
▼❛tr✐③❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦
❇r✉♥♦ ❈♦❡❧❤♦ ❆❧✈❡s
●♦✐â♥✐❛
TERMO DE CIÊNCIA E DE AUTORIZAÇÃO PARA DISPONIBILIZAR ELETRONICAMENTE OS TRABALHOS DE CONCLUSÃO DE CURSO NA BIBLIOTECA DIGITAL DA UFG
Na qualidade de titular dos direitos de autor, autorizo a Universidade Federal de Goiás (UFG) a disponibilizar, gratuitamente, por meio da Biblioteca Digital de Teses e Dissertações (BDTD/UFG), sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o do-cumento conforme permissões assinaladas abaixo, para fins de leitura, impressão e/ou down-load, a título de divulgação da produção científica brasileira, a partir desta data.
1. Identificação do material bibliográfico: Trabalho de Conclusão de Curso de Mestrado Profissional
2. Identificação do Trabalho
Autor (a): Bruno Coelho Alves E-mail: b.coelhoalves@gmail.com
Seu e-mail pode ser disponibilizado na página? [X]Sim [ ] Não Vínculo empregatício do autor Professor da UEG – Câmpus Santa Helena
Agência de fomento: - Sigla:
-País: - UF: - CNPJ:
-Título: Uma Proposta de Oficina sobre Códigos para a Contextualização do Estudo de Aritmética e Matrizes no Ensino Médio.
Palavras-chave: Códigos Corretores de Erros, Álgebra, Oficina para Ensino Médio. Título em outra língua: A Workshop Proposal on Codes for the contextualization of Arithmetic and Matrix Study in High School.
Palavras-chave em outra língua: Error-Correcting Codes, Algebra, Workshop for High School.
Área de concentração: Matemática do Ensino Básico. Data defesa: (dd/mm/aaaa) 07/08/2015
Programa de Pós-Graduação: PROFMAT – Mestrado Profissional em Matemática em Rede Nacional.
Orientador (a): Prof. Dr. Mário José de Souza. E-mail: mariojsouza@gmail.com Co-orientador(a):*
-E-mail:
-*Necessita do C PF quando não constar no SisPG
3. Informações de acesso ao documento:
Concorda com a liberação total do documento [ X ] SIM [ ] NÃO1
Havendo concordância com a disponibilização eletrônica, torna -se imprescindível o en-vio do(s) arquivo(s) em formato digital PDF ou DOC do trabalho de conclusão de curso.
O sistema da Biblioteca Digital de Teses e Dissertações garante aos auto res, que os ar-quivos contendo eletronicamente as teses, dissertações ou trabalhos de conclusão de curso, antes de sua disponibilização, receberão procedimentos de segurança, criptografia (para não permitir cópia e extração de conteúdo, permitindo apenas impressão fraca) usando o padrão do Acrobat.
________________________________________ Data: 07 / 08 / 2015 Assinatura do (a) autor (a)
1 Neste caso o documento será embargado por até um ano a partir da data de defesa. A ext ensão deste prazo
❇r✉♥♦ ❈♦❡❧❤♦ ❆❧✈❡s
❯♠❛ Pr♦♣♦st❛ ❞❡ ❖✜❝✐♥❛ s♦❜r❡ ❈ó❞✐❣♦s ♣❛r❛
❛ ❈♦♥t❡①t✉❛❧✐③❛çã♦ ❞♦ ❊st✉❞♦ ❞❡ ❆r✐t♠ét✐❝❛
❡ ▼❛tr✐③❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼ár✐♦ ❏♦sé ❞❡ ❙♦✉③❛✳
●♦✐â♥✐❛
Ficha catalográfica elaborada automaticamente
com os dados fornecidos pelo(a) autor(a), sob orientação do Sibi/UFG.
Alves, Bruno Coelho
Uma Proposta de Oficina sobre Códigos para a Contextualização do Estudo de Aritmética e Matrizes no Ensino Médio [manuscrito] / Bruno Coelho Alves. - 2015.
74 f.: il.
Orientador: Prof. Dr. Mário José de Souza.
Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Jataí, Programa de Pós-Graduação em Matemática (PROFMAT - Profissional), Goiânia, 2015.
Bibliografia. Apêndice. Inclui lista de figuras.
❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞♦ ♦r✐❡♥t❛❞♦r✳
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ à ❉❡✉s ♣♦r t✉❞♦ q✉❡ ✈✐✈✐ ❡♠ t♦❞♦s ❡ss❡s ❛♥♦s✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ♠ã❡ ❊❧✐③❛❜❡t❤ ❈✉♥❤❛ ❈♦❡❧❤♦ ❆❧✈❡s ✭■♥ ▼❡♠♦r✐❛♠✮✱ q✉❡ ❝♦♠♣❛r✲ t✐❧❤♦✉ ❞❡ss❡ s♦♥❤♦✱ s❡♥❞♦ ❢✉♥❞❛♠❡♥t❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ ❡✱ ✐♥❢❡❧✐③♠❡♥t❡✱ ♣❛rt✐♥❞♦ ❛♥t❡s ❞❛ ❝♦♥❝❧✉sã♦✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❡s♣♦s❛ ▼❛r✐❛♥♥❡ ❋❡rr❡✐r❛ ●♦♠❡s ❆❧✈❡s✱ q✉❡✱ ♣♦r ♠❛✐s ❞✐❢í❝✐❧ q✉❡ ❢♦✐✱ ❝♦♥s❡❣✉✐✉ ❝♦♠♣r❡❡♥❞❡r ❡ s✉♣♦rt❛r ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♣❛✐ ▲✉✐③ ❈❛r❧♦s ❆❧✈❡s✱ q✉❡ ❡♥t❡♥❞❡✉ ❡ ❛✉①✐❧✐♦✉ ✜♥❛♥❝❡✐r❛♠❡♥t❡ ❞✉r❛♥t❡ ❣r❛♥❞❡ ♣❛rt❡ ❞❡st❡✳
❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ✐r♠ã✱ r❡❢ú❣✐♦ ♣❛r❛ t♦❞❛ ❛ ♣r❡ssã♦ ❞❡ss❡s ♣♦✉❝♦ ♠❛✐s ❞❡ ❞♦✐s ❛♥♦s✳ ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼ár✐♦ ❏♦sé ❞❡ ❙♦✉③❛✱ s✉❛ ❝♦♥tr✐❜✉✐çã♦ ❢♦✐ ✐♥❡st✐♠á✈❡❧✱ ✐♥❝❡♥t✐✈♦✉ ❡ ❛✉①✐❧✐♦✉ ♥❡st❡ tr❛❜❛❧❤♦ ❛♣❡s❛r ❞❡ t♦❞♦s ♦s ❡♥tr❛✈❡s ♦❝♦rr✐❞♦s✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ♦ ❝♦♥t❡ú❞♦ ❞❡ ❈ó❞✐❣♦s ❈♦rr❡t♦r❡s ❞❡ ❊rr♦s ❛ ♣r♦❢❡ss♦r❡s✱ ❞❡ ♠♦❞♦ q✉❡ ♣♦ss❛♠ ✉t✐❧✐③á✲❧♦ ❡♠ s✉❛s ❛✉❧❛s✳ ❯♠ ❝ó❞✐❣♦ é ❛ r❡♣r❡✲ s❡♥t❛çã♦ ❞❡ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣❛❧❛✈r❛ ♦✉ sí♠❜♦❧♦ ♣♦r ✉♠❛ ♦✉tr❛ ♣❛❧❛✈r❛ ♦✉ sí♠❜♦❧♦✳ ❊st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛ ♦s ❝ó❞✐❣♦s ❝♦rr❡t♦r❡s ❞❡ ❡rr♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦s ❞✐t♦s ❧✐♥❡❛r❡s✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝ó❞✐❣♦ é ❛♣r❡s❡♥t❛❞♦ ❥✉♥t♦ ❛ ❞♦✐s ❡①❡♠♣❧♦s ♠♦t✐✈❛❞♦r❡s✳ ❖s r❡q✉✐s✐t♦s ❞❡ ➪❧❣❡❜r❛ ❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r sã♦ ❡①♣♦st♦s✱ ❛❜♦r❞❛♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝♦r♣♦s ✜♥✐t♦s✱ ❛ ❞❡✜♥✐çã♦ ❞❡ ❡s♣❛ç♦ ❧✐♥❡❛r ❡ ❞❡ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✳ ❖s ❝ó❞✐❣♦s ❝♦rr❡t♦✲ r❡s ❞❡ ❡rr♦s✱ ❛ ♣❛rt✐r ❞❡ ❛❧❢❛❜❡t♦s ❞❡✜♥✐❞♦s ❡♠ ❝♦r♣♦s ✜♥✐t♦s✱ ♣❡r♠✐t❡♠ q✉❡ ♦ ❡♥✈✐♦ ❞❡ ♠❡♥s❛❣❡♥s✱ ♠❡s♠♦ ❡♠ ❝❛♥❛✐s r✉✐❞♦s♦s✱ ♣♦ss❛♠ s❡r ✐♥t❡r♣r❡t❛❞❛s ❝♦♠ ✉♠❛ ♠❡♥♦r ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ ❡rr♦✳ ❊ss❡ ♣r♦❝❡ss♦ é ❢❡✐t♦ ❡♠ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s ❛tr❛✈és ❞❡ tr❛♥s❢♦r✲ ♠❛çõ❡s ❧✐♥❡❛r❡s✳ ❖s ❝ó❞✐❣♦s ❞❡ ❍❛♠♠✐♥❣ ❡ ♦s ❝ó❞✐❣♦s ❞❡ ❘❡❡❞✲❙♦❧♦♠♦♥ sã♦ ❡①❡♠♣❧♦s ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s q✉❡ sã♦ tr❛t❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳ ❆♣ós ❛♣r❡s❡♥t❛❞❛ ❛ ❜❛s❡ t❡ór✐❝❛ ❞♦ ❝♦♥t❡ú❞♦✱ é ♣r♦♣♦st❛ ✉♠❛ ♦✜❝✐♥❛ q✉❡ ♣♦ss✉✐ ❝♦♠♦ ❛❧✈♦ ♦s ❛❧✉♥♦s ❞♦s ❛♥♦s ✜♥❛✐s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❊ss❛ ♦✜❝✐♥❛ ❡①♣❧♦r❛ ❛❧❣✉♥s ❝ó❞✐❣♦s ❝♦♠✉♥s✱ ❝♦♠♦ ♦ ✉s♦ ❞❡ ❞í❣✐✲ t♦s ✈❡r✐✜❝❛❞♦r❡s ❡ ❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ t❡①t♦s ♣♦r ♠áq✉✐♥❛s ❞✐❣✐t❛✐s ❛tr❛✈és ❞❡ ③❡r♦s ❡ ✉♥s✳ ❊s♣❡r❛✲s❡ q✉❡ ❡st❡ tr❛❜❛❧❤♦ ♣♦ss❛ ❛✉①✐❧✐❛r ❛ ❞✐✈✉❧❣❛çã♦ ❞❡ ❛❧❣✉♥s tó♣✐❝♦s ❛t✉❛✐s ❞❡ ♣❡sq✉✐s❛ ❡♥tr❡ ♦s ♣r♦❢❡ss♦r❡s ❡ ✐♥❝❡♥t✐✈❛r ♦ ✉s♦ ❞❡ ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❡♥s✐✲ ♥❛r ❝♦♥t❡ú❞♦s q✉❡ sã♦ ❝♦♥s✐❞❡r❛❞♦s ✏❞✐❢í❝❡✐s✑ ❡ ✏✐♥út❡✐s✑ ♣❛r❛ ✈ár✐♦s ❛❧✉♥♦s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡
❆❜str❛❝t
❚❤✐s ✇♦r❦ ❛✐♠s t♦ ♣r❡s❡♥t t❤❡ ❝♦♥t❡♥t ♦❢ ❊rr♦r✲❈♦rr❡❝t✐♥❣ ❈♦❞❡s t♦ t❡❛❝❤❡rs✱ ✐♥ ❛ ✇❛② t♦ ✉s❡ t❤✐s ❝♦♥t❡♥t ✐♥ t❤❡✐r ❝❧❛ss❡s✳ ❆ ❝♦❞❡ r❡♣r❡s❡♥t✐♥❣ ❛ ♣❛rt✐❝✉❧❛r ✇♦r❞ ♦r s②♠❜♦❧ ❜② ❛♥♦t❤❡r ✇♦r❞ ♦r s②♠❜♦❧✳ ❚❤✐s st✉❞② ❤❛♥❞❧❡ ✇✐t❤ t❤❡ ❡rr♦r ❝♦rr❡❝t✐♥❣ ❝♦❞❡s✱ ❡s♣❡❝✐❛❧❧② t❤❛t s❛✐❞ ❧✐♥❡❛rs✳ ❚❤✉s✱ t❤❡ ❝♦♥❝❡♣t ♦❢ ❝♦❞❡ ✐s ❞✐s❝❧♦s❡❞ ❛❧♦♥❣ t✇♦ ❡①❛♠♣❧❡s✳ ❚❤❡ ▲✐♥❡❛r ❆❧❣❡❜r❛ ❛♥❞ ❆❧❣❡❜r❛ r❡q✉✐r❡♠❡♥ts ❛r❡ ❡①♣♦s❡❞✱ ❤❛♥❞❧✐♥❣ t❤❡ ❝♦♥❝❡♣t ♦❢ ✜♥✐t❡ ✜❡❧❞s✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❧✐♥❡❛r s♣❛❝❡ ❛♥❞ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥✳ ❚❤❡ ❡rr♦r ❝♦rr❡❝t✐♥❣ ❝♦❞❡s✱ ❢r♦♠ ❛❧♣❤❛❜❡ts ❞❡✜♥❡❞ ✐♥ ✜♥✐t❡ ✜❡❧❞s✱ ❛❧❧♦✇ s❡♥❞✐♥❣ ♠❡ss❛❣❡s✱ ❡✈❡♥ ✐♥ ♥♦✐s② ❝❤❛♥♥❡❧s✱ t❤❡② ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ✇✐t❤ ❛ ❧♦✇❡r ♣r♦❜❛❜✐❧✐t② ♦❢ ❡rr♦r✳ ❚❤✐s ♣r♦❝❡ss ✐s ❞♦♥❡ ✐♥ ❧✐♥❡❛r ❝♦❞❡s t❤r♦✉❣❤ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s✳ ❍❛♠♠✐♥❣ ❝♦❞❡s ❛♥❞ ❘❡❡❞✲❙♦❧♦♠♦♥ ❝♦❞❡s ❛r❡ ❡①❡♠♣❧❡s ♦❢ ❧✐♥❡❛r ❝♦❞❡s ✇❤✐❝❤ ❛r❡ s❤♦✇❡❞ ✐♥ t❤✐s ✇♦r❦✳ ❆❢t❡r ♣r❡s❡♥t❡❞ t❤❡ t❤❡♦r❡t✐❝❛❧ ❜❛s✐s ♦❢ ❝♦♥t❡♥t✱ ✐t ✐s ♣r♦♣♦s❡❞ ❛ ✇♦r❦s❤♦♣ t❤❛t ❤❛s ❛s t❛r❣❡t st✉❞❡♥ts ❢r♦♠ t❤❡ ✜♥❛❧ ②❡❛rs ♦❢ ❍✐❣❤ ❙❝❤♦♦❧✳ ❚❤✐s ✇♦r❦s❤♦♣ ❡①♣❧♦r❡ s♦♠❡ ❝♦♠♠♦♥ ❢❡❛t✉r❡ ❝♦❞❡s✱ s✉❝❤ ❛s t❤❡ ✉s❡ ♦❢ ❝❤❡❝❦ ❞✐❣✐ts ❛♥❞ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❡①ts ❜② ❞✐❣✐t❛❧ ♠❛❝❤✐♥❡s ❜② ③❡r♦s ❛♥❞ ♦♥❡s✳ ■t ✐s ❤♦♣❡❞ t❤❛t t❤✐s ✇♦r❦ ❝❛♥ ❛ss✐st t❤❡ r❡❧❡❛s❡ ♦❢ s♦♠❡ ❝✉rr❡♥t r❡s❡❛r❝❤ t♦♣✐❝s ❛♠♦♥❣ t❡❛❝❤❡rs ❛♥❞ ❡♥❝♦✉r❛❣❡ t❤❡ ✉s❡ ♦❢ ♥❡✇ ♠❡t❤♦❞♦❧♦❣✐❡s ❢♦r t❡❛❝❤✐♥❣ ❝♦♥t❡♥t t❤❛t ✐s ❝♦♥s✐❞❡r❡❞ ✏ ❞✐✣❝✉❧t ✑ ❛♥❞ ✏ ✉s❡❧❡ss ✑ t♦ s❡✈❡r❛❧ st✉❞❡♥ts ♦❢ ❍✐❣❤ ❙❝❤♦♦❧✳
❑❡②✇♦r❞s
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶✹
✶ ❈ó❞✐❣♦s ✶✻
✶✳✶ ❇❛s❡s ♥✉♠ér✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✷ ❖ ❝ó❞✐❣♦ ❆❙❈■■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷ ❈♦r♣♦s ❋✐♥✐t♦s ❡ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ✷✵ ✷✳✶ ❈♦r♣♦s ❋✐♥✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸ ❈ó❞✐❣♦s ❈♦rr❡t♦r❡s ❞❡ ❊rr♦s ✷✻ ✸✳✶ ❈❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ❡ ❝♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ■s♦♠❡tr✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸ ▼✉❞❛♥ç❛ ❞❡ ❆❧❢❛❜❡t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹ ❈ó❞✐❣♦s ▲✐♥❡❛r❡s ✸✼
✹✳✶ ❈♦♥❝❡✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✹✳✷ ▼❛tr✐③ ●❡r❛❞♦r❛ ❞❡ ✉♠ ❈ó❞✐❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✸ ❈ó❞✐❣♦s ❉✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✹ ❈♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦ ❝♦♠ ✉♠ ❝ó❞✐❣♦ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺ ❊①❡♠♣❧♦s ❞❡ ❈ó❞✐❣♦s ▲✐♥❡❛r❡s ✹✽ ✺✳✶ ❈ó❞✐❣♦s ❞❡ ❍❛♠♠✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✷ ❈ó❞✐❣♦s ❞❡ ❘❡❡❞✲❙♦❧♦♠♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✻ ❖✜❝✐♥❛ ✲ ■♥tr♦❞✉çã♦ ❛♦s ❝ó❞✐❣♦s ✺✶ ✻✳✶ ■♠♣♦rtâ♥❝✐❛ ❞❡ ✉♠❛ ♦✜❝✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
✻✳✷ ❆ ♦✜❝✐♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✻✺
❆ ❆t✐✈✐❞❛❞❡s ♣❛r❛ ❛ ✸❛ ❛✉❧❛ ✻✽
❇ ❆t✐✈✐❞❛❞❡s ♣❛r❛ ❛ ✼❛ ❛✉❧❛ ✻✾
❈ ❆t✐✈✐❞❛❞❡s ♣❛r❛ ❛ ✶✵❛ ❛✉❧❛ ✼✶
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✼✸
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ❊sq✉❡♠❛ ❞❡ ✉♠ ❝❛♥❛❧ ❝♦❞✐✜❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✻✳✶ ❈❤❡q✉❡ ❞♦ ❇❛♥❝♦ ❞♦ ❇r❛s✐❧ ❝♦♠ ❛❣ê♥❝✐❛ ❡ ❝♦♥t❛ ❞❡st❛❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✻✳✷ ❈❛❞❛str♦ ❞❡ P❡ss♦❛ ❋ís✐❝❛ ✭❈P❋✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ❇✳✶ ■♠❛❣❡♠ ♣❛r❛ ◗✉❡stã♦ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
■♥tr♦❞✉çã♦
❖s ❝ó❞✐❣♦s s❡ ❛♣r❡s❡♥t❛♠ ❞❡ ❢♦r♠❛s tã♦ s✐♠♣❧❡s ♥♦ ❝♦t✐❞✐❛♥♦ q✉❡✱ às ✈❡③❡s✱ ♥♦s ♣❡r♠✐t❡ ✐❣♥♦r❛r s❡✉ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❡ ✉s✉❢r✉✐r ❞❡ s❡✉s r❡s✉❧t❛❞♦s✳ ❉❡ss❛ ❢♦r♠❛✱ ❛❝❛❜❛✲ s❡ ♣♦r ♥ã♦ ♣❡r❝❡❜❡r ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s ❡♠ ❝♦♥t❛s ❜❛♥❝ár✐❛s✱ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❧❡✐t✉r❛ ❞❡ ❝ó❞✐❣♦s ❞❡ ❜❛rr❛✱ ❛ ♠❡❧❤♦r❛ ♥❛ q✉❛❧✐❞❛❞❡ ❞❛ r❡❝❡♣çã♦ ❞❡ ✐♠❛❣❡♥s ♣♦r ❝♦rr❡✐♦ ❡❧❡trô♥✐❝♦✱ ❡♥tr❡ ♦✉tr♦s✳
➱ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡r❛❧ q✉❡✱ ❝♦♠ ♦ ❛❞✈❡♥t♦ ❞♦ ❝♦♠♣✉t❛❞♦r ❡ ❞❛ ■♥t❡r♥❡t✱ q✉❡ ♦s ❞❛❞♦s ♣♦❞❡♠ s❡r tr❛♥s♠✐t✐❞♦s ❛♣❡♥❛s ✉s❛♥❞♦ ③❡r♦s ❡ ✉♥s✳ ❖ ♣r♦❝❡ss♦ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥♦ ❝♦♥t❡①t♦ ❛t✉❛❧✱ ❡①✐❣❡ q✉❡ ❛ r❡❝❡♣çã♦ s❡❥❛ ♦ ♠❛✐s ❝♦♥✜á✈❡❧ ♣♦ssí✈❡❧✱ ✐st♦ é✱ ❝♦♠ ❛ ♠❡♥♦r q✉❛♥t✐❞❛❞❡ ❞❡ ❡rr♦s✳ ❉❡t❡❝t❛r ❡rr♦s ♥❡ss❡ ♣r♦❝❡ss♦ ❡ ❝♦♥s❡❣✉✐r ❝♦rr✐❣í✲❧♦s é ✉♠❛ ❛t✐✈✐❞❛❞❡ ❡st✉❞❛❞❛ ❡ ❞❡s❡♥✈♦❧✈✐❞❛ ♣♦r ✉♠❛ ár❡❛ ❞❛ ▼❛t❡♠át✐❝❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❚❡♦r✐❛ ❞♦s ❈ó❞✐❣♦s✳
❆ ❚❡♦r✐❛ ❞♦s ❈ó❞✐❣♦s t❡♠ ❝♦♠♦ ✉♠ ❞❡ s❡✉s r❛♠♦s ❛ ❚❡♦r✐❛ ❞♦s ❈ó❞✐❣♦s ❈♦rr❡t♦r❡s ❞❡ ❊rr♦s✱ ❢✉♥❞❛❞❛ ❡♠ ✶✾✹✽ ♣❡❧♦ ▼❛t❡♠át✐❝♦ ❈✳ ❊✳ ❙❤❛♥♥♦♥✳ ■♥✐❝✐❛❧♠❡♥t❡ ❛♣❡♥❛s ❞❡ ✐♥t❡r❡ss❡ ❞♦s ♠❛t❡♠át✐❝♦s✱ ❞❡♥tr♦ ❞❛s ❞é❝❛❞❛s ❞❡ ✺✵ ❡ ✻✵✱ ♠♦str♦✉ s✉❛ ✉t✐❧✐❞❛❞❡ ❛ ♣❛rt✐r ❞❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ❞❡✈✐❞♦ ❛♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❝♦♠♣✉t❛❞♦r❡s ❡ ❞❛ ♥❡❝❡ss✐❞❛❞❡ ❞❛ tr❛♥s♠✐ssã♦ ❞❡ ❞❛❞♦s ❛tr❛✈és ❞❡ ❣r❛♥❞❡s ❞✐stâ♥❝✐❛s✳ ❯♠ ❡①❡♠♣❧♦ é ♦ ❡♥✈✐♦ ❞❡ ✐♠❛❣❡♥s ❞❛ ▲✉❛ ♣♦r s✐♥❛✐s ♥ã♦ ❛♥t❡s ♣♦ssí✈❡✐s✱ ❥á q✉❡ ❛ ✐♥t❡r❢❡rê♥❝✐❛ ❞❡ r❛✐♦s ♥ã♦ ♣❡r♠✐t✐❛♠ ❝❧❛r❡③❛ ♥❛ r❡❝❡♣çã♦✳ ❍♦❥❡✱ t❡♠♦s ❛ tr❛♥s♠✐ssã♦ ❞❡ s✐♥❛❧ ❞✐❣✐t❛❧ ❞❡ ❚❱ q✉❡ ♥❡❝❡ss✐t❛ ❞❡ss❡s ❝ó❞✐❣♦s ♣❛r❛ tr❛♥s♠✐ssõ❡s ✉s❛♥❞♦ ❛ t❡❝♥♦❧♦❣✐❛ ✹❑ ❞❡ ❢♦r♠❛ rá♣✐❞❛✱ ♣♦ss✐❜✐❧✐t❛♥❞♦ ✉♠❛ ♠❛✐♦r r♦❜✉st❡③ às ✐♥t❡r❢❡rê♥❝✐❛s ♣r❡s❡♥t❡s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❡♥✈✐♦ ❞♦ s✐♥❛❧✳
❊st❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ❢♦❝♦ ❞❡s❡♥✈♦❧✈❡r ♣❛rt❡ ❞❛ t❡♦r✐❛ ❞♦s ❝ó❞✐❣♦s ❝♦rr❡t♦r❡s ❞❡ ❡rr♦s✳ ❯t✐❧✐③❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❛♣r❡s❡♥t❛❞♦s✱ é ❞❡s❡♥✈♦❧✈✐❞♦ ✉♠❛ ♦✜❝✐♥❛✱ ❞❡st✐♥❛❞❛ ❛♦s ❛❧✉♥♦s ❞❡ ❊♥s✐♥♦ ▼é❞✐♦✱ ❝♦♠ ♦ ✐♥t✉✐t♦ ❞❡ ♠♦t✐✈❛r ♦ ❡♥s✐♥♦ ❞❡ ❞✐✈❡rs♦s t❡♠❛s q✉❡ ❝♦♠♣õ❡♠ ♦ ❝✉rrí❝✉❧♦ ❜ás✐❝♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
▼❛t❡♠át✐❝❛ ❡ ■♥❢♦r♠át✐❝❛✳ ➱ ❛♣r❡s❡♥t❛❞❛ ❛ ❝♦♥✈❡rsã♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣❛r❛ ♦✉tr❛s ❜❛s❡s ♥✉♠ér✐❝❛s ❡ ♦ ❝ó❞✐❣♦ ❆❙❈■■ ❝♦♠♦ ❡①❡♠♣❧♦ ❞♦ ✉s♦ ❝♦t✐❞✐❛♥♦ ❞❡ss❡ ❡st✉❞♦✳ ❆ ♠✉❞❛♥ç❛ ❞❡ ❜❛s❡ é ❢✉♥❞❛♠❡♥t❛❧ ♥❡ss❡ ♠♦♠❡♥t♦ ♣❛r❛ ♠♦str❛r ❝♦♠♦ é ♣♦ssí✈❡❧ ✐❞❡♥t✐✜❝❛r ❛tr❛✈és ❞❡ ❝♦r♣♦s ✜♥✐t♦s ❡❧❡♠❡♥t♦s ❞✐✈❡rs♦s✱ ❝♦♠♦ ♣❛❧❛✈r❛s ❡ ✐♠❛❣❡♥s✳
◆♦ ❈❛♣ít✉❧♦ ✷ é ❛♣r❡s❡♥t❛❞♦ ❝♦♥❝❡✐t♦s ♣r❡❧✐♠✐♥❛r❡s ❞❡ ➪❧❣❡❜r❛ q✉❡ ❢✉♥❞❛♠❡♥t❛♠ ♦s ❡st✉❞♦s ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s ❡ ❝ó❞✐❣♦s ❝í❝❧✐❝♦s✳ ❈♦♥❝❡✐t♦s ❞♦ ❡st✉❞♦ ❞❡ ♣♦❧✐♥ô♠✐♦s s♦❜r❡ ❝♦r♣♦s ✜♥✐t♦s ❡ ❞❡ ➪❧❣❡❜r❛ ▲✐♥❡❛r ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞❡ ❝ó❞✐❣♦s ❝♦rr❡t♦r❡s ❞❡ ❡rr♦s sã♦ ❛♣r❡s❡♥t❛❞♦s ❡ ❞✐s❝✉t✐❞♦s✳
❈♦♠ ❜❛s❡ ♥♦s t❡♠❛s ♣r❡❧✐♠✐♥❛r❡s ♠♦str❛❞♦s✱ ♦ ❈❛♣ít✉❧♦ ✸ ❡①♣õ❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❝ó❞✐❣♦s ❝♦rr❡t♦r❡s ❞❡ ❡rr♦s✱ s✉❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s✱ ❝♦♠♦ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ ❡ ❡①✐stê♥❝✐❛ ❞❡ ✐s♦♠❡tr✐❛s✱ ❡ ❡①❡♠♣❧♦s✳
❖ ❈❛♣ít✉❧♦ ✹ tr❛③ ❛ ❝❧❛ss❡ ❞♦s ❝ó❞✐❣♦s ❧✐♥❡❛r❡s✱ q✉❡ sã♦ ♦s ♠❛✐s ✉s❛❞♦s ♥♦ ❝♦t✐❞✐❛♥♦ ❞❡✈✐❞♦ às s✉❛s ❝❛r❛❝t❡ríst✐❝❛s✳ ❙✉❛ ❞❡✜♥✐çã♦✱ ❛❧❣✉♠❛s ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦ ♣r♦✲ ❝❡ss♦ ❞❡ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡s❝♦❞✐✜❝❛çã♦ ❡♥✈♦❧✈✐❞♦ ♥❡ss❡ ❡st✉❞♦ ❡stã♦ ❛♣r❡s❡♥t❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦✳
❖ ❈❛♣ít✉❧♦ ✺ ❛♣r❡s❡♥t❛ ♦s ❝ó❞✐❣♦s ❞❡ ❍❛♠♠✐♥❣ ❡ ♦s ❝ó❞✐❣♦s ❞❡ ❘❡❡❞✲❙♦❧♦♠♦♥✱ ❡①❡♠♣❧♦s ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s ❞❡ ❢á❝✐❧ ✐♠♣❧❡♠❡♥t❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦✳
P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✻ ❡♥❝♦♥tr❛✲s❡ ♦ ❡sq✉❡♠❛ ❞❡ ✉♠❛ ♦✜❝✐♥❛ ♣❛r❛ ✐❧✉str❛r ❞❡ ❢♦r♠❛ s✐♠♣❧❡s ♦s t❡♠❛s ❞❡s❡♥✈♦❧✈✐❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳ Pr❡t❡♥❞❡✲s❡ t❛♠❜é♠ ✐♥st✐❣❛r ♦s ❛❧✉♥♦s ❛ r❡❛❧✐③❛çã♦ ❞❡ ♣❡sq✉✐s❛s✳
❆♣r❡s❡♥t❛r ❛ ▼❛t❡♠át✐❝❛ ❝♦♠♦ ✉♠ s❡r ✈✐✈♦ ❡♠ ❝♦♥st❛♥t❡ ❡✈♦❧✉çã♦ é ❞❡✈❡r ❞❡ t♦❞♦ ♣r♦❢❡ss♦r✳ ❊ss❡ ❢❛t♦ ♥♦rt❡✐❛ ❛ ❡①✐stê♥❝✐❛ ❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❖ ♣r♦❢❡ss♦r ♣r❡❝✐s❛ s❡ ♠❛♥t❡r ❛t✉❛❧✐③❛❞♦ ❡ ❛❜❡rt♦ ❛♦s ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✱ ♣r✐♥❝✐✲ ♣❛❧♠❡♥t❡ ❛q✉❡❧❡s q✉❡ s❡ r❡❢❡r❡♠ ❛ t❡♠❛s ❞❛ ❛t✉❛❧✐❞❛❞❡✱ ✉♠❛ ✈❡③ q✉❡ ♦s ❛❧✉♥♦s✱ ❞✉r❛♥t❡ ♦ ❝♦♥t❛t♦ ❝♦♠ ❛ ✉t✐❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♦❜t✐❞♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❡stã♦ ♠❛✐s ♣r♦♣í✲ ❝✐♦s ❛ ❛❜s♦r✈❡r ❝♦♠ ♠❛✐s q✉❛❧✐❞❛❞❡ ✉♠ ❝♦♥t❡ú❞♦✳ ❆❧❣✉♥s ❞♦s ❝♦♥t❡ú❞♦s ❛q✉✐ ❡①♣♦st♦s✱ ❝♦♠♦ ♣♦❧✐♥ô♠✐♦s ❡ ♠❛tr✐③❡s✱ sã♦ ♦❜❥❡t♦s ❞❡ ✉♠❛ ♣❡r❣✉♥t❛ ♠✉✐t♦ ❝♦♠✉♠ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✿ ✏P❛r❛ q✉❡ s❡r✈❡ ✐ss♦❄✑✳ ●r❛♥❞❡ ♣❛rt❡ ❞♦s ♣r♦❢❡ss♦r❡s ♥ã♦ ❝♦♥s❡❣✉❡♠ r❡s♣♦♥❞❡r ♣r♦♥t❛♠❡♥t❡ ❡ss❡ q✉❡st✐♦♥❛♠❡♥t♦✳ ❊st❡ tr❛❜❛❧❤♦ ✈✐s❛ ♣♦ss✐❜✐❧✐t❛r ♦ ❛❝❡ss♦ ❛♦s ♣r♦❢❡s✲ s♦r❡s ❞❡ ✉♠❛ ❢♦r♠❛ ❞❡ ❛❧t❡r❛r ❡ss❛ r❡❛❧✐❞❛❞❡✱ ❛❧é♠ ❞❡ ❝♦♥t❡①t✉❛❧✐③❛r ❛❧❣✉♥s ❝♦♥t❡ú❞♦s✱ ♥❡❝❡ss✐❞❛❞❡ ❡①♣♦st❛ ❡♠ ❞✐✈❡rs♦s ♣❛râ♠❡tr♦s ❝✉rr✐❝✉❧❛r❡s ♣r♦♣♦st♦s ♣❡❧♦ ❣♦✈❡r♥♦✳
❈❛♣ít✉❧♦ ✶
❈ó❞✐❣♦s
❯♠ ❝ó❞✐❣♦ é ✉♠ s✐st❡♠❛ ❞❡ ♣❛❧❛✈r❛s ♦✉ ♦✉tr♦s sí♠❜♦❧♦s ✉s❛❞♦s ♣❛r❛ r❡♣r❡s❡♥t❛r ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣❛❧❛✈r❛s ♦✉ ♦✉tr♦s sí♠❜♦❧♦s✳ ➱ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❡ss❡ ❝♦♥❝❡✐t♦ é ❝♦♠✉♠ ❛♦s s✐st❡♠❛s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ♠✉✐t♦ ✈✐st♦s ❡♠ ✜❧♠❡s ❞❡ ❡s♣✐♦♥❛❣❡♠ ❡ ❧✐✈r♦s✱ ♠❛s ❡❧❡ é ❛♣❧✐❝❛❞♦ ❛ ❝ó❞✐❣♦s ♠❛✐s ✏❝♦♠✉♥s✑ ❛♦ ❝♦t✐❞✐❛♥♦✱ t❛✐s ❝♦♠♦ ❝ó❞✐❣♦s ❞❡ ❜❛rr❛s✱ qr ❝♦❞❡s ❡ ♦ s✐st❡♠❛ ❇r❛✐❧❧❡✳ ❊ss❡ ❝❛♣ít✉❧♦ ❡①♣õ❡ ♦ t❡♠❛ ❝♦♠♦ ❡①♣❧♦r❛❞♦ ❡♠ ❬✷❪✳
✶✳✶ ❇❛s❡s ♥✉♠ér✐❝❛s
❯♠ ❡①❡♠♣❧♦ ❞❡ ❝ó❞✐❣♦ é ❛ ❡s❝r✐t❛ ❞❡ ♥ú♠❡r♦s ❡♠ ♦✉tr❛s ❜❛s❡s ♥✉♠ér✐❝❛s✳
❖ s✐st❡♠❛ ♥✉♠ér✐❝♦ ♠❛✐s ✉s❛❞♦ ♥♦ ❝♦t✐❞✐❛♥♦ é ♦ s✐st❡♠❛ ❞❡❝✐♠❛❧✳ ❊ss❡ s✐st❡♠❛ ❝♦♥s✐st❡ ❡♠ ❥✉st❛♣♦r ❛❧❣❛r✐s♠♦s✱ ♦♥❞❡ ❝❛❞❛ ❛❧❣❛r✐s♠♦ ♣♦ss✉✐ ✉♠ ✈❛❧♦r q✉❡ ❞❡♣❡♥❞❡ ❞❛ s✉❛ ♣♦s✐çã♦✳ ❯♠ ♥ú♠❡r♦ ❞❡❝✐♠❛❧ ❞❛ ❢♦r♠❛
m=anan−1an−2· · ·a1a0
♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦
m=an·10n+an−1·10n−1+an−2·10n−2+· · ·+a1·101+a0·100 =
n
X
i=0 ai10i.
❊①✐st❡♠ ♦✉tr❛s ❜❛s❡s q✉❡ ❛♣r❡s❡♥t❛♠ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ♣♦r ♣♦ss✉ír❡♠ ✉♠ ♥ú♠❡r♦ ♠❡♥♦r ❞❡ ❛❧❣❛r✐s♠♦s✱ ♠✐♥✐♠✐③❛♥❞♦ ♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡rr♦s✱ ♦✉ ♣♦r ❛♣r❡s❡♥t❛r❡♠ ♠❛✐s ❛❧❣❛r✐s♠♦s✱ ♣❡r♠✐t✐♥❞♦ ❛ ❡s❝r✐t❛ ❞❡ ♠❛✐s ✐♥❢♦r♠❛çõ❡s ❡♠ ✉♠ ♠❡♥♦r ❡s♣❛ç♦✳ ❊♥tr❡
❡❧❛s é ♣♦ssí✈❡❧ ❝✐t❛r ❛s ❜❛s❡s ❜✐♥ár✐❛✭❝♦♠♣♦st❛ ♣♦r ❞♦✐s ❛❧❣❛r✐s♠♦s✱ ✵ ❡ ✶✮ ❡ ❛ ❜❛s❡ ❤❡①❛❞❡❝✐♠❛❧✭❝♦♠♣♦st❛ ♣❡❧♦s ❛❧❣❛r✐s♠♦s ❞❛ ❜❛s❡ ❞❡❝✐♠❛❧✱ ❛❝r❡s❝✐❞♦s ❞❛s ❧❡tr❛s ❆✱ ❇✱ ❈✱ ❉✱ ❊✱ ❋✮✳ ❆ ✜♠ ❞❡ ♠✐♥✐♠✐③❛r ♣♦ssí✈❡❧ ❝♦♥❢✉sã♦✱ ♦s ♥ú♠❡r♦s ♥❛ ❜❛s❡ ❜✐♥ár✐❛ ❡stã♦ ❛q✉✐ r❡♣r❡s❡♥t❛❞♦s ♥❛ ❢♦r♠❛(x)2 ❡ ♥❛ ❜❛s❡ ❤❡①❛❞❡❝✐♠❛❧ ♥❛ ❢♦r♠❛(x)16✳
❯♠ ♥ú♠❡r♦ ❜✐♥ár✐♦ ♣♦ss✉✐ r❡♣r❡s❡♥t❛çã♦ ❞❡❝✐♠❛❧ s❡ ❡s❝r✐t♦ ❞❛ ❢♦r♠❛
(bnbn−1· · ·b2b1b0)2 =bn·2n+bn−1·2n−1+· · ·+b2·22 +b1·21+b0·20 =
n
X
i=0 bi2i.
❯♠ ♥ú♠❡r♦ ❤❡①❛❞❡❝✐♠❛❧ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♥❛ ❢♦r♠❛ ❞❡❝✐♠❛❧ ❝❛s♦ s❡❥❛ ❡s❝r✐t♦ ❝♦♠♦
(hnhn−1· · ·h2h1h0)16=hn·16n+hn−1·16n−1+· · ·+h2·162+h1·161+h0·160 =
n
X
i=0 hi16i,
♦♥❞❡ ♦s ❛❧❣❛r✐s♠♦s ❆✱ ❇✱ ❈✱ ❉✱ ❊✱ ❋ ❞❡✈❡♠ s❡r s✉❜st✐t✉í❞♦s ♣♦r ✶✵✱ ✶✶✱ ✶✷✱ ✶✸✱ ✶✹✱ ✶✺✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
➱ ♣♦ssí✈❡❧ r❡♣r❡s❡♥t❛r ♥ú♠❡r♦s ❡♠ ✉♠❛ ❜❛s❡ ✐♥t❡✐r❛ ♣♦s✐t✐✈❛ q✉❛❧q✉❡r✱ s❡❥❛ ♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♦✉ ♥ã♦✳ ❊ss❡ ♣r♦❝❡ss♦ é ❢❡✐t♦ ✉t✐❧✐③❛♥❞♦ ✉♠ ♣r♦❝❡ss♦ ❞❡ ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s✳ ❊①❡♠♣❧♦ ✶✳ ❈♦♥✈❡rt❡r ♦ ♥ú♠❡r♦(2102)3 ♣❛r❛ ❛ ❜❛s❡ ❜✐♥ár✐❛✳
❘❡s♦❧✉çã♦✿ ❙❛❜❡✲s❡ q✉❡ (2102)3 = 2·33 + 1·32 + 0·31+ 2·30 = 65 ❡♠ ❞❡❝✐♠❛✐s✳
❋❛③❡♥❞♦ ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s✱ t❡♠✲s❡
65 = 2·32 + 1
32 = 2·16 + 0
16 = 2·8 + 0
8 = 2·4 + 0
4 = 2·2 + 0
2 = 2·1 + 0
1 = 2·0 + 1
❉❡ss❛ ❢♦r♠❛✱ t♦♠❛♥❞♦ ♦s r❡st♦s ❝♦♠♦ ❝♦❡✜❝✐❡♥t❡s ❞❛s ♣♦tê♥❝✐❛s✱ t❡♠✲s❡ q✉❡(2102)3 =
65 = 1·26+ 0·25 + 0·24+ 0·23+ 0·22+ 0·21+ 1·20 = (1000001) 2✳
P♦❞❡ s❡r ♠❡♥❝✐♦♥❛❞♦ q✉❡ ♦ ♣r✐♥❝✐♣❛❧ ♠♦t✐✈❛❞♦r ❞♦ ✉s♦ ❞❡ ♥ú♠❡r♦s ❜✐♥ár✐♦s ❡ ❤❡✲ ①❛❞❡❝✐♠❛✐s ♥♦ ❝♦t✐❞✐❛♥♦ s❡ r❡❢❡r❡ ❛♦ ❛❞✈❡♥t♦ ❞❛ ✐♥❢♦r♠át✐❝❛✳ ❈♦♠♣✉t❛❞♦r❡s ❞✐❣✐t❛✐s
tr❛❜❛❧❤❛♠ ✐♥t❡r♥❛♠❡♥t❡ ❝♦♠ ♥ú♠❡r♦s ❜✐♥ár✐♦s ❡ ♣❡r♠✐t❡♠ ❡♥tr❛❞❛s ✉s❛♥❞♦ ❝ó❞✐❣♦s ❤❡①❛❞❡❝✐♠❛✐s ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ r❡❢❡rê♥❝✐❛✳ ❈ó❞✐❣♦s ❛♥t✐❣♦s✱ ❝♦♠♦ ♦ ❝ó❞✐❣♦ ♠♦rs❡✱ ❛♣r❡✲ s❡♥t❛♠ r❡♣r❡s❡♥t❛çã♦ ❡♠ ❜❛s❡ t❡r♥ár✐❛✱ ✐st♦ é✱ ✉t✐❧✐③❛♥❞♦ três ❛❧❣❛r✐s♠♦s✳
✶✳✷ ❖ ❝ó❞✐❣♦ ❆❙❈■■
❖ ❝ó❞✐❣♦ ❆❙❈■■✭❆♠❡r✐❝❛♥ ❙t❛♥❞❛rt ❈♦❞❡ ❢♦r ■♥❢♦r♠❛t✐♦♥ ■♥t❡r❝❤❛♥❣❡ ✲ ❈ó❞✐❣♦ P❛❞rã♦ ❆♠❡r✐❝❛♥♦ ♣❛r❛ ■♥t❡r❝â♠❜✐♦ ❞❡ ■♥❢♦r♠❛çã♦✮ é ✉♠ ❝ó❞✐❣♦ ❜✐♥ár✐♦ q✉❡ ♣♦ss✐❜✐❧✐t❛ ❛ ❝♦♥✈❡rsã♦ ❞❡ ♥ú♠❡r♦s ♣❛r❛ ❧❡tr❛s ♠❛✐ús❝✉❧❛s ❡ ♠✐♥ús❝✉❧❛s✱ ❛❧é♠ ❞❡ ❛❧❣✉♥s sí♠❜♦❧♦s ❡ ❝❛r❛❝t❡r❡s ❞❡ ❝♦♥tr♦❧❡ q✉❡ ❛❧t❡r❛♠ ♦ ♣r♦❝❡ss❛♠❡♥t♦ ❞♦ t❡①t♦ ✐♥s❡r✐❞♦✳
❆ ❚❛❜❡❧❛ ✶✳✶✱ ❡①tr❛í❞❛ ❞❡ ❬✷❪✱ ❛♣r❡s❡♥t❛ ♦s ✈❛❧♦r❡s ❞♦ ❝ó❞✐❣♦ ❆❙❈■■ ❞❛❞❛ ❡♥tr❛❞❛ ❤❡①❛❞❡❝✐♠❛❧✱ ✐st♦ é✱ ❞❡✈❡ s❡r r❡❛❧✐③❛❞❛ ❛ ♦♣❡r❛çã♦ ❞❡ ❛❞✐çã♦ ❡♥tr❡ ♦s ✈❛❧♦r❡s ❞❛s ❧✐♥❤❛s ❡ ❞❛s ❝♦❧✉♥❛s s❡❣✉✐♥❞♦ ❛ ♥✉♠❡r❛çã♦ ❤❡①❛❞❡❝✐♠❛❧✳❖s sí♠❜♦❧♦s ♣r❡s❡♥t❡s ♥♦ ✐♥t❡✈❛❧♦ ❡♥tr❡ ✵✵ ❡ ✶❋✱ ✐st♦ é✱ ❛s q✉❛tr♦ ♣r✐♠❡✐r❛s ❧✐♥❤❛s ❞❛ ❚❛❜❡❧❛ ✶✳✶✱ sã♦ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❝❛r❛❝t❡r❡s ❞❡ ❝♦♥tr♦❧❡✱ ❡❧❡s ❛❧t❡r❛♠ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞♦❝✉♠❡♥t♦ ❡ ♥ã♦ sã♦ ✐♠♣r❡ss♦s ♥♦ r❡s✉❧t❛❞♦✳ ❆ ♣❛rt✐r ❞❡ss❡ ❝ó❞✐❣♦✱ é ♣♦ssí✈❡❧ ❝♦❞✐✜❝❛r ♣❛❧❛✈r❛s ❛tr❛✈és ❞❡ ❝♦♠❛♥❞♦ ❤❡①❛❞❡❝✐♠❛✐s ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❡♠ ✈❛❧♦r❡s ❜✐♥ár✐♦s✱ q✉❡ ♣❡r♠✐t❡♠ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ✉♠ t❡①t♦ ♣♦r ✉♠❛ ♠áq✉✐♥❛ ❞✐❣✐t❛❧✳
❊①❡♠♣❧♦ ✷✳ ❈♦♥✈❡rt❡r ❛ ♣❛❧❛✈r❛ ✏❆❧✉♥♦✑ ♣❛r❛ ❤❡①❛❞❡❝✐♠❛❧ ❡✱ ❡♠ s❡❣✉✐❞❛✱ ❡♠ ❜✐♥ár✐♦✱ ✉s❛♥❞♦ ❛ t❛❜❡❧❛ ❆❙❈■■✳
❘❡s♦❧✉çã♦✿ ❯s❛♥❞♦ ❛ ❚❛❜❡❧❛ ✶✳✶✱ ♦❜té♠✲s❡ ❆✿✹✶ ▲✿✻❈ ❯✿✼✺ ◆✿✻❊ ❖✿✻❋✱ q✉❡ ♣♦ss✉✐ r❡♣r❡s❡♥t❛çã♦ ❜✐♥ár✐❛ ✵✶✵✵✵✵✵✶ ✵✶✶✵✶✶✵✵ ✵✶✶✶✵✶✵✶ ✵✶✶✵✶✶✶✵ ✵✶✶✵✶✶✶✶✱ ♦♥❞❡ ♦s ③❡r♦s à ❡sq✉❡r❞❛ ❢♦r❛♠ ❛❞✐❝✐♦♥❛❞♦s ♣❛r❛ q✉❡ ❝❛❞❛ r❡♣r❡s❡♥t❛çã♦ t❡♥❤❛ ✽ ❛❧❣❛r✐s♠♦s ✭❞✐t♦s ❜✐ts ❡♠ ❧✐♥❣✉❛❣❡♠ ✐♥❢♦r♠❛❝✐♦♥❛❧ ❡ ❝❛❞❛ ✉♠ ❞❡ss❡s ❝♦♥❥✉♥t♦s ❞❡ ✽ ❜✐ts é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❜②t❡✮✳
❖ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r ♠♦str❛ ❝♦♠♦ é ♣♦ssí✈❡❧ ♦ ♣r♦❝❡ss♦ ❞❡ ❝♦❞✐✜❝❛çã♦ ❞❡ ♣❛❧❛✈r❛s✳ ❊①✐st❡♠ ♦✉tr♦s ❝ó❞✐❣♦s q✉❡ ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s s❡♠❡❧❤❛♥t❡s à t❛❜❡❧❛ ❆❙❈■■✱ ❝♦♠♦ ♦s ❝ó❞✐❣♦s ❞❡ ❜❛rr❛s ❡ ♦s qr ❝♦❞❡s✱ q✉❡ sã♦ r❡♣r❡s❡♥t❛♥t❡s ❜✐❞✐♠❡♥s✐♦♥❛✐s ❞♦ ♣r♦❝❡ss♦ ❞❡ ❝♦❞✐✜❝❛çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s✳
▼❡✐♦s ✉s❛❞♦s ♣❛r❛ tr❛♥s♣♦rt❛r ✐♥❢♦r♠❛çõ❡s ❡♥tr❡ ♦ ❡♠✐ssár✐♦ ❡ ♦ r❡❝❡♣t♦r sã♦ ❞❡♥♦✲ ♠✐♥❛❞♦s ❝❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çã♦✳ ❊ss❡ ♣r♦❝❡ss♦ ❡♥✈♦❧✈❡ ♦ ✉s♦ ❞❡ ❝♦❞✐✜❝❛çõ❡s ❡ ❞❡❝♦❞✐✲ ✜❝❛çõ❡s ❞❡ ♠❡♥s❛❣❡♥s✱ ❛❧é♠ ❞❡ s❡ t❡♥t❛r ❡✈✐t❛r ♦ ♠á①✐♠♦ ♣♦ssí✈❡❧ ❞❡ r✉í❞♦s✳ ❖ ❡st✉❞♦ ❞❡ss❡ ♣r♦❝❡ss♦ é ❛❜♦r❞❛❞♦ ♥♦ ❝❛♣ít✉❧♦ s❡❣✉✐♥t❡✳
✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✵✵ ◆❯▲ ❙❖❍ ❙❚❳ ❊❚❳ ❊❖❚ ❊◆◗ ❆❈❑ ❇❊▲ ✵✽ ❇❙ ❍❚ ▲❋ ❱❚ ❋❋ ❈❘ ❙❖ ❙■ ✶✵ ❉▲❊ ❉❈✶ ❉❈✷ ❉❈✸ ❉❈✹ ◆❆❑ ❙❨◆ ❊❚❇ ✶✽ ❈❆◆ ❊▼ ❙❯❇ ❊❙❈ ❋❙ ●❙ ❘❙ ❯❙ ✷✵ ✦ ✑ ★ ✩ ✪ ✫ ✬ ✷✽ ✭ ✮ ✯ ✰ ✱ ✲ ✳ ✴ ✸✵ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✸✽ ✽ ✾ ✿ ❀ ❁ ❂ ❃ ❄ ✹✵ ❅ ❆ ❇ ❈ ❉ ❊ ❋ ● ✹✽ ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ ✺✵ P ◗ ❘ ❙ ❚ ❯ ❱ ❲ ✺✽ ❳ ❨ ❩ [ \ ] ✂ ❴
✻✵ ❵ ❛ ❜ ❝ ❞ ❡ ❢ ❣ ✻✽ ❤ ✐ ❥ ❦ ❧ ♠ ♥ ♦ ✼✵ ♣ q r s t ✉ ✈ ✇ ✼✽ ① ② ③ { ⑤ } ✄ ❉❊▲
❚❛❜❡❧❛ ✶✳✶✿ ❝ó❞✐❣♦ ❆❙❈■■
❈❛♣ít✉❧♦ ✷
❈♦r♣♦s ❋✐♥✐t♦s ❡ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s
❖s ❝♦r♣♦s ✜♥✐t♦s ❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧✐♥❡❛r❡s sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✲ ✈✐♠❡♥t♦ ❞♦ ❡st✉❞♦ ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s✳ ❆ ❜❛s❡ t❡ór✐❝❛ ❞❡st❡ ❝❛♣ít✉❧♦ ❢✉♥❞❛♠❡♥t❛✲s❡ ❡♠ ❬✹❪✱ ❬✻❪✱ ❬✼❪ ❡ ❬✶✵❪✳
✷✳✶ ❈♦r♣♦s ❋✐♥✐t♦s
❯♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ é ✉♠ ❝♦♥❥✉♥t♦ R ♠✉♥✐❞♦ ❞❡ ❞✉❛s ♦♣❡r❛çõ❡s✿
+ : R×R −→ R e · : R×R −→ R
(a, b) 7−→ a+b (a, b) 7−→ a·b
❝❤❛♠❛❞❛s ❞❡ ❛❞✐çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦ss✉✐♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡✲ ❞❛❞❡s✿
❉❛❞♦sa, b, c,0,1∈R✱
✶✳ ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦✮ a+ (b+c) = (a+b) +c;
✷✳ ✭❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❛ ❛❞✐çã♦✮a+ 0 = 0 +a=a;
✸✳ ✭❡❧❡♠❡♥t♦ ✐♥✈❡rs♦ ❞❛ ❛❞✐çã♦✮ a+ (−a) = (−a) +a= 0;
✹✳ ✭❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ❛❞✐çã♦✮ a+b=b+a;
✺✳ ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮(a·b)·c=a·(b·c);
✻✳ ✭❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮a·1 = 1·a=a;
✼✳ ✭❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✮ a·b =b·a;
✽✳ ✭❞✐str✐❜✉t✐✈✐❞❛❞❡ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ r❡❧❛çã♦ à ❛❞✐çã♦✮ a·(b+c) = a·b+a·c.
❖ ❡❧❡♠❡♥t♦ a+b é ❞✐t♦ s♦♠❛ ❡ ♦ ❡❧❡♠❡♥t♦ a·b✱ t❛♠❜é♠ ❡s❝r✐t♦ ❝♦♠♦ ab✱ é ❞✐t♦
♣r♦❞✉t♦✳
❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠ ❛♥❡❧ R é ❞✐t♦ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✱ s❡ ❢♦r ✈á❧✐❞♦ ∀a, b∈A, a6= 0 eb 6= 0⇒a·b 6= 0.
❊①❡♠♣❧♦s ❞❡ ❞♦♠í♥✐♦s ❞❡ ✐♥t❡❣r✐❞❛❞❡ sã♦ ♦s ❛♥é✐sZ✱ Q✱R❡C✳ ❊♠ ✉♠ ❞♦♠í♥✐♦ ❞❡
✐♥t❡❣r✐❞❛❞❡ é ✈á❧✐❞❛ ❛ ❧❡✐ ❞❡ ❝❛♥❝❡❧❛♠❡♥t♦ ♣❛r❛ ✐❣✉❛❧❞❛❞❡s ❞❡ ♣r♦❞✉t♦s✳
❉❡✜♥✐çã♦ ✷✳✶✳✷✳ ❯♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❛♥❡❧R é ❝❤❛♠❛❞♦ ✐♥✈❡rtí✈❡❧ s❡a·a−1 = 1✱ ♦♥❞❡
♦ ✐♥✈❡rs♦ ❞❡ a é ♦ ❡❧❡♠❡♥t♦ a−1✳
❯♠ ❛♥❡❧ ❡♠ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦✱ ❞✐❢❡r❡♥t❡ ❞♦ ✵✱ s❡❥❛ ✐♥✈❡rtí✈❡❧✱ é ❞❡♥♦♠✐♥❛❞♦ ❝♦r♣♦✳ ❉❡✜♥✐çã♦ ✷✳✶✳✸✳ ❯♠ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ ❡ ♥ã♦ ✐♥✈❡rtí✈❡❧a ❞❡ ✉♠ ❛♥❡❧ R é ❞✐t♦ ♣r✐♠♦
s❡
∀b, c∈R, a|b·c⇒a|boua|c.
❖♥❞❡ a|b s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x∈R t❛❧ q✉❡ ax=b✳
❉❡✜♥✐çã♦ ✷✳✶✳✹✳ ❙❡❥❛ R ✉♠ ❛♥❡❧ ❡ m∈R✳ ❉❛❞♦s ❡❧❡♠❡♥t♦s a, b∈R✱ ❞✐r❡♠♦s q✉❡a
é ❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ m✱ ❡ s❡ ❡s❝r❡✈❡
a≡bmodm
❝❛s♦ m|(a−b)✳
❆s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ❡♥✈♦❧✈❡♥❞♦ ❝♦♥❣r✉ê♥❝✐❛s sã♦ ✈á❧✐❞❛s✿ ❉❛❞♦sa, b, c, a′, b′ ∈R✱ ❡♥tã♦
✶✳ a ≡amodm❀
✷✳ s❡ a≡bmodm✱ ❡♥tã♦ b≡amodm❀
✸✳ s❡ a≡bmodm ❡ b ≡cmodm✱ ❡♥tã♦ a≡cmodm❀
✹✳ s❡a ≡a′modm❡b ≡b′modm✱ ❡♥tã♦a+b≡a′+b′modm ❡a·b≡a′·b′modm✳
❆s três ♣r✐♠❡✐r❛s ♣r♦♣r✐❡❞❛❞❡s ❝❛r❛❝t❡r✐③❛♠ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❝♦♠♦ ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳
❉❡✜♥✐çã♦ ✷✳✶✳✺✳ ❆ ❝❧❛ss❡ r❡s✐❞✉❛❧ ❞❡ ✉♠ ❡❧❡♠❡♥t♦a ∈R✱ ♠ó❞✉❧♦ m✱ é ♦ ❝♦♥❥✉♥t♦
[a] ={x∈R;x≡amodm}={a+m·λ;λ∈R}
❖ ❛♥❡❧ Rm✱ ❢♦r♠❛❞♦ ♣❡❧❛s ❝❧❛ss❡s r❡s✐❞✉❛✐s ♠ó❞✉❧♦ m✱ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛♥❡❧ ❞♦s
✐♥t❡✐r♦s ♠ó❞✉❧♦m✳ ❈❛s♦ms❡❥❛ ✉♠ ❡❧❡♠❡♥t♦ ♣r✐♠♦p✱ ♦ ❛♥❡❧Rpé ✉♠ ❝♦r♣♦✱ ❝♦♠✉♠❡♥t❡
❞❡♥♦♠✐♥❛❞♦ ❝♦r♣♦ ❞❡ ●❛❧♦✐s✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r GF(p)✳ ❈♦♠♦ ♦ ❝♦♥❥✉♥t♦ Rp é ✜♥✐t♦✱
❡ss❡ ❝♦r♣♦ é ❞❡♥♦♠✐♥❛❞♦ ❝♦♠♦ ❝♦r♣♦ ✜♥✐t♦✳ ❖s ❝♦r♣♦s ✜♥✐t♦s ❡stã♦ r❡♣r❡s❡♥t❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦ ♣♦rFp✳
P♦❧✐♥ô♠✐♦s ♣♦❞❡♠ s❡r ❝♦♥str✉í❞♦s ❛ ♣❛rt✐r ❞❡ ❝♦r♣♦s ✜♥✐t♦s✳ ❯♠ ♣♦❧✐♥ô♠✐♦ é ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ F[x]✱ ♦♥❞❡ ♦s ❝♦❡✜❝✐❡♥t❡s sã♦ ❡❧❡♠❡♥t♦s ❞♦ ❝♦r♣♦ ✜♥✐t♦ F s♦❜r❡
✉♠❛ ✈❛r✐á✈❡❧x✳ ❆ss✐♠✱ ❡ss❡ ❝♦♥❥✉♥t♦ é ❞❡s❝r✐t♦ ❝♦♠♦✿
F[x] =
( n X
i=0
ci·xi =c0x0+c1x1+c2x2+c3x3+· · ·+cnxn
)
♦♥❞❡ci ∈F, i= 1,2, . . . , n✳
❆s ♦♣❡r❛çõ❡s ❡♥tr❡ ♦s ♣♦❧✐♥ô♠✐♦s sã♦ ❛s ✉s✉❛✐s✱ ❝♦♠ ❛ s♦♠❛ ❡ ♣r♦❞✉t♦ ❡♥tr❡ ♦s ❝♦❡✜❝✐❡♥t❡s r❡❛❧✐③❛❞❛s ❞❡♥tr♦ ❞❡F✳
✷✳✷ ❊s♣❛ç♦s ❱❡t♦r✐❛✐s
❉❡✜♥✐çã♦ ✷✳✷✳✶✳ ❉❛❞♦ ✉♠ ❝♦r♣♦F✱ V é ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ s♦❜r❡ ✉♠ ❝♦r♣♦Fs❡ ♣♦ss✉✐
✉♠❛ ❛❞✐çã♦ ❡♥tr❡ ❡❧❡♠❡♥t♦s ✉,✈,✇∈V✱ ♠✉♥✐❞❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s✿ • é ❛ss♦❝✐❛t✐✈❛✱ ✐st♦ é✱ (✉+✈) +✇=✉+ (✈+✇)❀
• é ❝♦♠✉t❛t✐✈❛✱ ✐st♦ é✱ ✉+✈=✈+✉❀
• ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ ✐st♦ é✱ ✈+✵ =✵+✈ =✈❀
• ♣♦ss✉✐ s✐♠étr✐❝♦s✱ ✐st♦ é✱ ✈+ (✲✈) =✵✳
❊✱ ❛❧é♠ ❞✐ss♦✱ ❡①✐st❡ ✉♠❛ ♦♣❡r❛çã♦ ❞❡♥♦♠✐♥❛❞❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡s❝❛❧❛r✱ q✉❡ ❛ss♦❝✐❛ ✉♠ ❡❧❡♠❡♥t♦ a∈F ❛ ✉♠ ❡❧❡♠❡♥t♦ ✈∈V✱ ✉♠ ❡❧❡♠❡♥t♦ a✈ ∈V✱ t❛❧ q✉❡
• é ❞✐str✐❜✉t✐✈❛✱ ✐st♦ é✱ a(✉+✈) = a✉+a✈ ❡ (a+b)✈=a✈+b✈❀ • ❛ss♦❝✐❛t✐✈❛ ❡♠ r❡❧❛çã♦ ❛♦ ♣r♦❞✉t♦ ❡s❝❛❧❛r✱ ✐st♦ é✱ (ab)✈ =a(b✈)❀
• ♣♦ss✉✐ ❡❧❡♠❡♥t♦ ♥❡✉tr♦✱ ✐st♦ é✱ 1✈ =✈✳
❖s ❡❧❡♠❡♥t♦s ❞❡V sã♦ ❞✐t♦s ✈❡t♦r❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ F ❞❡ ❡s❝❛❧❛r❡s✳
❊①❡♠♣❧♦ ✸✳ ❖s ♣♦❧✐♥ô♠✐♦s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥♦ ❝♦r♣♦ ✜♥✐t♦F✱ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ F[x]✱
❝♦♠ ❛s ♦♣❡r❛çõ❡s ✉s✉❛✐s✱ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳
❯♠ s✉❜❡s♣❛ç♦ ✈❡t♦r✐❛❧ é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ q✉❡ ♠❛♥té♠ ❛s ♠❡s♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳
❉❡✜♥✐çã♦ ✷✳✷✳✷✳ ❙❡❥❛♠ ✈1,✈2,· · · ,✈n ✈❡t♦r❡s ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V✳ ❉✐③✲s❡ q✉❡
❡ss❡s ✈❡t♦r❡s sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❝❛s♦ ❛ ❡q✉❛çã♦
a1✈1+a2✈2 +· · ·+an✈n= 0
é s❛t✐s❢❡✐t❛ ❛♣❡♥❛s q✉❛♥❞♦ a1 = a2 =· · · =an = 0✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❡ss❡s ✈❡t♦r❡s sã♦
❞✐t♦s ❧✐♥❡❛r♠❡♥t❡ ❞❡♣❡♥❞❡♥t❡s✳
❖ ❝♦♥❝❡✐t♦ ❞❡ ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s é út✐❧ ♥❛ ❞❡✜♥✐çã♦ ❞❡ ❜❛s❡ ❡ ❞✐✲ ♠❡♥sã♦ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧✳
❉❡✜♥✐çã♦ ✷✳✷✳✸✳ ❙❡❥❛α✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣♦st♦ ❡①❝❧✉s✐✈❛♠❡♥t❡ ♣♦r ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡
✐♥❞❡♣❡♥❞❡♥t❡s✳ ❉✐③❡♠♦s q✉❡α é ✉♠❛ ❜❛s❡ ❞❡ ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ V s❡ t♦❞♦ ✈❡t♦r ✈ ❞❡ V ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ✈❡t♦r❡s ❞♦ ❝♦♥❥✉♥t♦ α✱ ✐st♦ é✱
✈=a1✈1+a2✈2+· · ·+an✈n,
♦♥❞❡ ✈1,✈2,· · · ,✈n ∈V✳
❖ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❜❛s❡ é ❞✐t♦ ❞✐♠❡♥sã♦ ❞❡ V ❡ ❞❡♥♦t❛❞♦ ♣♦rdim V✳
◆♦ ❊①❡♠♣❧♦ ✸✱ ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ ❡s♣❛ç♦ ❞♦s ♣♦❧✐♥ô♠✐♦s é ♦ ❝♦♥❥✉♥t♦ ❞♦s ✈❡t♦r❡s
1, x, x2,· · · , xn✱ ❝✉❥❛ ❞✐♠❡♥sã♦ é n+ 1✳
❆❧❣✉♠❛s ❢✉♥çõ❡s ❡♥tr❡ ❡s♣❛ç♦s ❧✐♥❡❛r❡s ♣♦ss✉❡♠ ♣r♦♣r✐❡❞❛❞❡s ❡s♣❡❝í✜❝❛s✱ ♦ q✉❡ ❡①✐❣❡ ✉♠❛ ❞❡✜♥✐çã♦ ♣ró♣r✐❛✳
❉❡✜♥✐çã♦ ✷✳✷✳✹✳ ❙❡❥❛♠ V ❡ W ❡s♣❛ç♦s ✈❡t♦r✐❛✐s✳ ❯♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r ❞❡ V
❡♠ W é ✉♠❛ ❢✉♥çã♦ T :V →W q✉❡ ♣♦ss✉✐ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✶✳ T(✈1+✈2) = T(✈1) +T(✈2)✱ ♣❛r❛ q✉❛✐sq✉❡r ✈1 ❡ ✈2 ❡♠ V❀
✷✳ T(a✈) =aT(✈)✱ ♣❛r❛ q✉❛✐sq✉❡r ✈ ❡♠ ❱ ❡ a ❡♠ F.
❊①❡♠♣❧♦ ✹✳ ❆ ❛♣❧✐❝❛çã♦ T(x, y) = (x+y, x−y,2x,2y) é ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r
❞♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ R2 ♥♦ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ R4✱ ♣♦✐s✱ ❞❛❞♦s ✉ = (x1, y1) ❡ ✈ = (x2, y2)
✈❡t♦r❡s ❞❡ R2 ❡ ✉♠ ❡s❝❛❧❛r α∈R✿
• T(✉+✈) =T(x1+x2, y1+y2) = (x1+x2+y1+y2, x1+x2−y1−y2,2x1+2x2,2y1+
2y2) = (x1+y1, x1−y1,2x1,2y1) + (x2+y2, x2−y2,2x2,2y2) = T(✉) +T(✈); • T(α✉) = T(αx1, αy1) = (αx1 +αy1, αx1 −αy1,2αx1,2αx2) = α(x1 +y1, x1 −
y1,2x1,2x2) =αT(✉)
❖ ❝♦♥❝❡✐t♦ ❞❡ ♥ú❝❧❡♦ ❡ ✐♠❛❣❡♠ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✳
❉❡✜♥✐çã♦ ✷✳✷✳✺✳ ❖ ♥ú❝❧❡♦ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛rT✱ ❞❡✜♥✐❞❛ ♣♦r T :V →W✱
❞❡♥♦t❛❞♦ ♣♦r Ker T✱ é ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❡t♦r❡s ❞❡ V q✉❡ ♣♦ss✉❡♠ ♦ ✈❡t♦r ♥✉❧♦ ❞❡ W
❝♦♠♦ ✐♠❛❣❡♠✱ ✐st♦ é✱
Ker T ={✈∈V;T(✈) = 0}.
❆ ✐♠❛❣❡♠ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r T é ♦ ❝♦♥❥✉♥t♦
Im T =T(V) = {✇∈W;∃✈∈V ⇒T(✈) =✇}.
➱ ✐♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ ♦ ♥ú❝❧❡♦ ❡ ❛ ✐♠❛❣❡♠ ❞❡ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r sã♦ s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡V ❡ W✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♠ r❡s✉❧t❛❞♦ ❞❡❝♦rr❡♥t❡ ❞❡ss❛ ❞❡✜✲
♥✐çã♦✱ ❞❡ ✉s♦ ♥♦ ❡st✉❞♦ ❞❡ ❝ó❞✐❣♦s ❧✐♥❡❛r❡s✱ é ♠♦str❛❞♦ ❛ s❡❣✉✐r✳
❚❡♦r❡♠❛ ✷✳✷✳✶✳ ❙❡❥❛ T : V → W ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❧✐♥❡❛r✱ ♦♥❞❡ V t❡♠ ❞✐♠❡♥sã♦
✜♥✐t❛✳ ❊♥tã♦
dim Ker T +dim Im T =dim V
❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ dim V = n✱ ❡ ✉♠❛ ❜❛s❡ B ❞❡ Ker T ❝♦♠ m ✈❡t♦r❡s✳ ➱
❝❧❛r♦ q✉❡m≤n✱ ♣♦✐s ❡①✐st❡♠ ♥♦ ♠á①✐♠♦ n ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❡♠ V✳
❆ss✉♠❡✲s❡ ❞♦✐s ❝❛s♦s✿
✐✮ m = n✿ ◆❡ss❡ ❝❛s♦✱ dim Ker T = dim V✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ❛ ❜❛s❡ ❞❡ Ker T
t❛♠❜é♠ é ✉♠❛ ❜❛s❡ ❞❡V✱ ✐♠♣❧✐❝❛♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ ❡♥tr❡ ❡ss❡s ❝♦♥❥✉♥t♦s✳ ❉❛í✱ Im T ={0} ⇒dim Im T = 0,
♠♦str❛♥❞♦ ❛ ✈❛❧✐❞❛❞❡ ❞❛ ❢ór♠✉❧❛✳
✐✐✮ m < n✿ ◆❡ss❡ ❝❛s♦✱ ♣♦❞❡♠♦s ❝♦♠♣❧❡t❛r ❛ ❜❛s❡ B ❞❡ Ker T ❛té ♦❜t❡r ✉♠❛
❜❛s❡ ♣❛r❛ V✱ ✐♥s❡r✐♥❞♦ n − m ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s ❛♦s ❥á ❡①✐st❡♥t❡s
❡♠ B✳ ❙❡❥❛ B′ = {✈m+1,✈m+2, . . . ,✈n} ♦s ✈❡t♦r❡s ❛❞✐❝✐♦♥❛❞♦s✳ ❯♠❛ ✈❡③ q✉❡ B′ é
✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ V✱ ♣♦✐s é ❢♦r♠❛❞♦ ♣♦r ✈❡t♦r❡s ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥✲
t❡s✱ ❡♥tã♦ {T(✈m+1), T(✈m+2), . . . , T(✈n)} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ Im T✳ ❉❛í✱
dim Im T =n−m✱ s❛t✐s❢❛③❡♥❞♦ ♦ ❚❡♦r❡♠❛✳
▼❛✐♦r❡s ❞❡t❛❧❤❡s ❞❡ss❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡♠ s❡r ♦❜t✐❞♦s ❡♠ ❬✹❪✳
❈❛♣ít✉❧♦ ✸
❈ó❞✐❣♦s ❈♦rr❡t♦r❡s ❞❡ ❊rr♦s
❊st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ♦ ♣r♦❝❡ss♦ ❞❡ tr❛♥s♠✐ssã♦ ❞❡ ♠❡♥s❛❣❡♥s ❝♦❞✐✜❝❛❞❛s✱ t❡♥❞♦ ❝♦♠♦ ❢♦❝♦ ❛ ❞❡t❡❝çã♦ ❡ ❛ ♣♦ssí✈❡❧ ❝♦rr❡çã♦ ❞❡ ❡rr♦s✳ ❆ ❜❛s❡ t❡ór✐❝❛ ❞❡st❡ ❝❛♣ít✉❧♦ s❡❣✉❡ ♦ ❡①♣♦st♦ ❡♠ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❬✺❪ ❡ ❬✽❪✳
✸✳✶ ❈❛♥❛✐s ❞❡ ❝♦♠✉♥✐❝❛çã♦ ❡ ❝♦❞✐✜❝❛çã♦
❯♠ ❝❛♥❛❧ ❞❡ ❝♦♠✉♥✐❝❛çã♦ é ✉♠ ♣r♦❝❡ss♦ q✉❡ ❡♥✈♦❧✈❡ ♦ ❡♥✈✐♦ ❞❡ ✉♠❛ ✐♥❢♦r♠❛çã♦ ❡♥✲ tr❡ ❞♦✐s ❡♥t❡s✱ ❞❡♥♦♠✐♥❛❞♦s ❡♠✐ss♦r✭❛q✉❡❧❡ q✉❡ ❡♥✈✐❛ ❛ ✐♥❢♦r♠❛çã♦✮ ❡ r❡❝❡♣t♦r✭❛q✉❡❧❡ q✉❡ r❡❝❡❜❡ ❛ ✐♥❢♦r♠❛çã♦✮✳ ❉✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ tr❛♥s♠✐ssã♦ ❞❛ ♠❡♥s❛❣❡♠ ❡♠ ❝❛✲ ♥❛✐s r✉✐❞♦s♦s ❡①✐st❡ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❛ ❛❧t❡r❛çã♦ ❞❛ ♠❡s♠❛✳ ❘✉í❞♦ é q✉❛❧q✉❡r ♠❡✐♦ q✉❡ ❛❧t❡r❡ ❛ ♠❡♥s❛❣❡♠ ❞✉r❛♥t❡ ♦ ❡♥✈✐♦✳ ❊❧❡ ♣♦❞❡ ♦❝♦rr❡r ❞❡✈✐❞♦ ✉♠❛ ❢♦♥t❡ tér♠✐❝❛✱ ❡❧étr✐❝❛✱ ❤✉♠❛♥❛✱ ✐♠♣❡r❢❡✐çõ❡s ♥♦ ❡q✉✐♣❛♠❡♥t♦✱ ❡t❝✳ ❆ss✐♠✱ ♥❡ss❡s ❝❛s♦s ❡①✐st❡ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ❛ ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛ ❞✐✜r❛ ❞❛ ♠❡♥s❛❣❡♠ ❡♥✈✐❛❞❛✳
❖ ♦❜❥❡t✐✈♦ ❞❡ ✉♠ ❝ó❞✐❣♦ ❝♦rr❡t♦r ❞❡ ❡rr♦s é ❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠ ❞❡ ❢♦r♠❛ ❛ ❡✈✐t❛r ❛ ❛❧t❡r❛çã♦ ❞❛ ♠❡♥s❛❣❡♠ ♣♦r r✉í❞♦s✱ ❞❡ ❢♦r♠❛ ❛ ❝♦rr✐❣✐r ♣r♦✈á✈❡✐s ❡rr♦s✱ ❞❡s❞❡ q✉❡ ❡♠ ♥ú♠❡r♦ ♥ã♦ ❛❧é♠ ❛♦ ♣♦❞❡r ❞❡ ❝♦rr❡çã♦ ❞♦ ❝ó❞✐❣♦ ✉t✐❧✐③❛❞♦✳
P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✸❪✱ ♣♦❞❡✲s❡ ♣❡♥s❛r ♥♦ ❡♥✈✐♦ ❞❡ ✉♠❛ ♠❡♥✲ s❛❣❡♠ à ✉♠ r♦❜ô q✉❡ ❞❡t❡r♠✐♥❛ s✉❛ ❞✐r❡çã♦✳ ❖s ❝♦♠❛♥❞♦ ♥♦rt❡✱ s✉❧✱ ❧❡st❡ ❡ ♦❡st❡ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❛❜❛✐①♦✳
norte7−→10 sul 7−→11
leste 7−→00 oeste7−→01
❖ ❝ó❞✐❣♦ à ❞✐r❡✐t❛ é ❝❤❛♠❛❞♦ ❞❡ ❝ó❞✐❣♦ ❢♦♥t❡✳ ❙✉♣♦♥❞♦ q✉❡ ♦ ❝❛♥❛❧ ♣♦r ♦♥❞❡ é ❡♥✈✐❛❞♦ ♦ ❝ó❞✐❣♦ ❢♦♥t❡ ❡①✐st❛ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ r✉í❞♦✱ é ❢❡✐t❛ ✉♠❛ ♥♦✈❛ ❝♦❞✐✜❝❛çã♦✱ ❛❞✐❝✐♦♥❛♥❞♦ ❛♦ ❝ó❞✐❣♦ ❢♦♥t❡ ✉♠❛ s❡q✉ê♥❝✐❛ r❡❞✉♥❞❛♥t❡ ❞❡ ✈❛❧♦r❡s q✉❡ ♣❡r♠✐t❛♠ r❡❛✲ ❧✐③❛r ❛ ❞❡t❡❝çã♦ ❡ ❝♦rr❡çã♦ ❞❡ ❡rr♦s✳ ➱ ❝❤❛♠❛❞♦ ❞❡ ❝ó❞✐❣♦ ❞❡ ❝❛♥❛❧ ❡ss❡ ♥♦✈♦ ❝ó❞✐❣♦ ❣❡r❛❞♦✳ ❖ ❝ó❞✐❣♦ ❢♦♥t❡ ❛♣r❡s❡♥t❛❞♦ ❛❝✐♠❛ ♣♦❞❡ s❡r r❡❝♦❞✐✜❝❛❞♦ ❝♦♠♦✿
norte7−→10110 sul 7−→11101
leste7−→00000 oeste7−→01011
❆ss✐♠✱ ✉♠❛ ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛ ❝♦♠♦ 01010 ♣❡❧♦ r♦❜ô ♥❛ ✈❡r❞❛❞❡ ❞❡✈❡r✐❛ s❡r 01011✱
✉♠❛ ✈❡③ q✉❡ ❡ss❛ é ❛ ♠❛✐s ♣ró①✐♠❛ ❞❛ ♠❡♥s❛❣❡♠ r❡❝❡❜✐❞❛✱ ❡ ♦ r♦❜ô ❞❡✈❡rá r❡❛❧✐③❛r ♦ ❝♦♠❛♥❞♦ ✏♦❡st❡✑✳
❖ ♣r♦❝❡ss♦ ❡①♣♦st♦ ♣♦❞❡ s❡r ❡sq✉❡♠❛t✐③❛❞♦ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✸✳✶✳
❋✐❣✉r❛ ✸✳✶✿ ❊sq✉❡♠❛ ❞❡ ✉♠ ❝❛♥❛❧ ❝♦❞✐✜❝❛❞♦
❊st❡ tr❛❜❛❧❤♦ ❛❜♦r❞❛ ❛♣❡♥❛s ❝❛♥❛✐s s✐♠étr✐❝♦s✱ ✐st♦ é✱ ❛q✉❡❧❡s ❝✉❥❛ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ r❡❝❡❜✐♠❡♥t♦ ❞❡ sí♠❜♦❧♦s ❡rr❛❞♦s é ❡q✉✐♣r♦✈á✈❡❧ ❡✱ ❞❡♥tr❡ ♦s sí♠❜♦❧♦s ❡rr❛❞♦s ❛ s❡r❡♠ r❡❝❡❜✐❞♦s✱ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ r❡❝❡❜❡r q✉❛❧q✉❡r ✉♠ t❛♠❜é♠ é ❛ ♠❡s♠❛✳
❯♠ ❝ó❞✐❣♦ ❝♦rr❡t♦r ❞❡ ❡rr♦s é ✉♠ s✉❜❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ❞❡ An✱ ♦♥❞❡ A é ✉♠
❝♦♥❥✉♥t♦ ✜♥✐t♦✱ ❞❡♥♦♠✐♥❛❞♦ ❛❧❢❛❜❡t♦✱ q✉❡ ❝♦♥té♠ |A| = q ❡❧❡♠❡♥t♦s✱ ❞✐t♦ q−ár✐♦✱ ❡ n é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❯♠❛ ♣❛❧❛✈r❛ é ♦ ♥♦♠❡ ❞❛❞♦ ❛ ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝ó❞✐❣♦
❝♦rr❡t♦r ❞❡ ❡rr♦s✳ ◆♦ ❡①❡♠♣❧♦ ❞♦ r♦❜ô✱ ♦ ❝♦♥❥✉♥t♦A ={0,1}é ♦ ❛❧❢❛❜❡t♦✱ ❝♦♠ q= 2✱
❡ ❝❛❞❛ ♣❛❧❛✈r❛ ♣♦ss✉✐n= 5 ❞í❣✐t♦s✳
P❛r❛ q✉❡ s❡ ♣♦ss❛ ❝r✐❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r♦①✐♠✐❞❛❞❡ ❡♥tr❡ ♣❛❧❛✈r❛s é ♥❡❝❡ssár✐♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♠étr✐❝❛✱ ❝✉❥❛ ❞❡✜♥✐çã♦ s❡ ❡♥❝♦♥tr❛ ❛❜❛✐①♦✳
❉❡✜♥✐çã♦ ✸✳✶✳✶✳ ❉❛❞❛s ❞✉❛s ♣❛❧❛✈r❛s ✉✱ ✈∈An✱ ❛ ♠étr✐❝❛ ❞❡ ❍❛♠♠✐♥❣✱ ♦✉ ❞✐stâ♥❝✐❛
❞❡ ❍❛♠♠✐♥❣ ❡♥tr❡ ✉ ❡ ✈ é ❞❡✜♥✐❞❛ ❝♦♠♦
d(✉,✈) =|{i;ui 6=vi,1≤i≤n}|.
P♦r ❡①❡♠♣❧♦✱ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❛s ♣❛❧❛✈r❛s ✏♥♦rt❡✑ ❡ ✏s✉❧✑ ❞♦ ❝ó❞✐❣♦ ❞♦ r♦❜ô é
d(10110,11101) = 3.
Pr♦♣♦s✐çã♦ ✸✳✶✳✶✳ ❉❛❞♦s ✉,✈,✇∈An✱ ✈❛❧❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✐✮ P♦s✐t✐✈✐❞❛❞❡✿ d(✉,✈)≥0✱ ✈❛❧❡♥❞♦ ❛ ✐❣✉❛❧❞❛❞❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ✉=✈;
✐✐✮ ❙✐♠❡tr✐❛✿ d(✉,✈) = d(✈,✉);
✐✐✐✮ ❉❡s✐❣✉❛❧❞❛❞❡ tr✐❛♥❣✉❧❛r✿ d(✉,✈)≤d(✉,✇) +d(✇,✈).
❉❡♠♦♥str❛çã♦✿ ✐✮ ➱ ✐♠❡❞✐❛t❛ ❞❛ ❞❡✜♥✐çã♦✱ ♣♦✐s s❡ ❛ ✐❣✉❛❧❞❛❞❡ ♦❝♦rr❡r✱ ♥ã♦ ❡①✐st❡ ❝♦♠♣♦♥❡♥t❡s ❞✐st✐♥t❛s✱ ❧♦❣♦ ❛ ❞✐stâ♥❝✐❛ é ♥✉❧❛ ❡✱ ❝❛s♦ ❤❛❥❛ ❛❧❣✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❞✐❢❡✲ r❡♥t❡✱ ❛ ❞✐stâ♥❝✐❛ é ♣♦s✐t✐✈❛✳
✐✐✮ ❈♦♠♦ ui 6=vi ⇔vi 6=ui✱ s❡❣✉❡ ♦ r❡s✉❧t❛❞♦
✐✐✐✮ ❈♦♥s✐❞❡r❛✲s❡ ❞♦✐s ❝❛s♦s✿ ♥♦ ♣r✐♠❡✐r♦✱ s❡ ✉=✈✱ ❛ ❞✐stâ♥❝✐❛ é ♥✉❧❛✱ ❞❡ss❛ ❢♦r♠❛✱ ❛
❞✐stâ♥❝✐❛ ❞❡♥tr❡ ❡ss❛s ♣❛❧❛✈r❛s ❝♦♠ ✇ é ♥ã♦✲♥❡❣❛t✐✈❛✱ ❞❡✈✐❞♦ à ✐✮❀ ♥♦ s❡❣✉♥❞♦✱ ❝♦♥s✐❞❡r❡ ✉6=✈✱ ❛ ❝♦♥tr✐❜✉✐çã♦ ❞❛s ✐✲és✐♠❛s ❝♦♦r❞❡♥❛❞❛s à s♦♠❛ ❞❛s ❞✐stâ♥❝✐❛s é ✐❣✉❛❧ ❛ ✵✱ ✶ ♦✉
✷✱ ✐❣✉❛❧ ♦✉ ♠❛✐♦r q✉❡ ❛ ❝♦♥tr✐❜✉✐çã♦ ❡♥tr❡ ✉ ❡ ✈✱ q✉❡ sã♦ ✈❛❧♦r❡s ❡♥tr❡ ✵ ❡ ✶ ♣❛r❛ ❝❛❞❛ ❝♦♦r❞❡♥❛❞❛✳
❉❡✜♥✐çã♦ ✸✳✶✳✷✳ ❉❛❞♦ ✉♠ ❡❧❡♠❡♥t♦ ❛∈An ❡ ✉♠ ♥ú♠❡r♦ r❡❛❧t≥0✱ ❞❡✜♥✐✲s❡ ♦ ❞✐s❝♦
❡ ❛ ❡s❢❡r❛ ❞❡ r❛✐♦ t ❡ ❝❡♥tr♦ ❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦s ❝♦♥❥✉♥t♦s D(❛, t) = {✉∈An;d(✉,❛)≤t},
S(❛, t) ={✉∈An;d(✉,❛) =t},
❊ss❡s ❝♦♥❥✉♥t♦s sã♦ ✜♥✐t♦s✱ ❝♦♠ ❝❛r❞✐♥❛❧✐❞❛❞❡s ❞❛❞❛s ♣♦r
|S(❛, t)|=
n
t
(q−1)t,
|D(❛, t)|=
t
X
i=0
n t
(q−1)i.
❉❡✜♥✐çã♦ ✸✳✶✳✸✳ ❙❡❥❛ ❈ ✉♠ ❝ó❞✐❣♦✱ ❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ ❞❡ ❈ é
d=min{d(✉,✈);✉,✈∈C e✉6=✈}
❉❡ss❛ ❢♦r♠❛✱ é ♥❡❝❡ssár✐♦ ✉♠ ✉s♦ ❞❡ M
2
✱ ♦♥❞❡ M =|C|✱ ❝á❧❝✉❧♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s
♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛✱ ✉♠ ❝✉st♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♠✉✐t♦ ❛❧t♦ s❡ ♦ ✈❛❧♦r ❞❡
M ❢♦r ❣r❛♥❞❡✳
▲❡♠❛ ✸✳✶✳✶✳ ❙❡❥❛ C ✉♠ ❝ó❞✐❣♦ ❞❡ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ d✳ ❙❡ ❝,❝✬ ∈ C✱ ❝♦♠ c 6= c′✱
❡♥tã♦D(❝, κ)∩D(❝✬, κ) =∅.
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ ♣♦r ❛❜s✉r❞♦ q✉❡ ❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❞✐s❝♦s s❡❥❛ ♥ã♦ ✈❛③✐❛✱ ♦✉ s❡❥❛✱ q✉❡ ❡①✐st❛ ✈ t❛❧ q✉❡ ✈∈D(❝, κ)∩D(❝✬, κ)✱ ❞❛í
d(❝,❝✬)≤d(❝,✈) +d(❝✬,✈)≤κ+κ≤d−1≤d
♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐s d é ❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛✳
❆ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❞❡t❡r♠✐♥❛r ❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ ❞❡ ✉♠ ❝ó❞✐❣♦ é ❡①❡♠♣❧✐✜❝❛❞❛ ♥♦ ❚❡♦r❡♠❛ ✸✳✶✳✶✳
❚❡♦r❡♠❛ ✸✳✶✳✶✳ ❙❡❥❛ C ✉♠ ❝ó❞✐❣♦ ❝♦♠ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ d✳ ❊♥tã♦ C ♣♦❞❡ ❝♦rr✐❣✐r
❛té κ=
d−1
2
❡rr♦s ❡ ❞❡t❡❝t❛r ❛té d−1 ❡rr♦s✳
❉❡♠♦♥str❛çã♦✿ ❙❡ ❞✉r❛♥t❡ ❛ tr❛♥s♠✐ssã♦ ❞❡ ✉♠❛ ♣❛❧❛✈r❛ ❝ ❞♦ ❝ó❞✐❣♦ ❢♦r ❝♦♠❡t✐❞♦t
❡rr♦s✱ ❝♦♠t ≤κ✱ r❡❝❡❜❡♥❞♦ ❛ ♣❛❧❛✈r❛ r✱ ❡♥tã♦d(r,❝) = t≤κ❀ s❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ r ❛
q✉❛❧q✉❡r ♦✉tr❛ ♣❛❧❛✈r❛ ❞♦ ❝ó❞✐❣♦ é ♠❛✐♦r ❞♦ q✉❡ κ✱ ♣♦✐s D(❝, κ)∩D(❝✬, κ) = ∅✱ ♦♥❞❡
❝6=❝✬✳ ■ss♦ ❞❡t❡r♠✐♥❛ ❝ ✉♥✐✈♦❝❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ r✳
❉❡ss❛ ❢♦r♠❛✱ é ♣♦ssí✈❡❧ ❛ ❞❡t❡❝çã♦ ❞❡ ❡rr♦s ❝❛s♦ ❛ q✉❛♥t✐❞❛❞❡ s❡❥❛ ❞❡ ❛té d −1
❡rr♦s✱ ♠❛s é ♣♦ssí✈❡❧ ❝♦rr✐❣✐r ❛♣❡♥❛sκ ❞❡ss❡s ❡rr♦s✳
❉❡✜♥✐çã♦ ✸✳✶✳✹✳ ❯♠ ❝ó❞✐❣♦ C ⊂ An é ❞✐t♦ ♣❡r❢❡✐t♦✱ ❝♦♠ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛ d✱ ❡
♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❝♦rr❡çã♦ κ s❡
[
❝∈C
D(❝, κ) =An.
❆ ❉❡✜♥✐çã♦ ✸✳✶✳✹ ❣❛r❛♥t❡ q✉❡ ✉♠ ❝ó❞✐❣♦ ♣❡r❢❡✐t♦ ♣❡r♠✐t❡ ❛ ❝♦rr❡çã♦ ❞❡ q✉❛❧q✉❡r ❡rr♦ ❝♦♠❡t✐❞♦✳
❊ss❡ ♣r♦❝❡ss♦ ♣❡r♠✐t❡ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❡str❛té❣✐❛ ♣❛r❛ ❝♦rr❡çã♦ ❞❛ ♠❡♥s❛❣❡♠ r❡❝❡✲ ❜✐❞❛ ♣❡❧♦ r❡❝❡♣t♦r✿
✶✳ ❙❡ ❛ ♣❛❧❛✈r❛ r s❡ ❡♥❝♦♥tr❛ ❡♠ ✉♠ ❞✐s❝♦ ❞❡ r❛✐♦ κ✱ s✉❜st✐t✉✐✲s❡ r ♣♦r ❝✱ ♦♥❞❡ ❝ é
♦ ❝❡♥tr♦ ❞♦ ❞✐s❝♦✳
✷✳ ❙❡ ❛ ♣❛❧❛✈r❛ r ♥ã♦ s❡ ❡♥❝♦♥tr❛ ❡♠ ♥❡♥❤✉♠ ❞✐s❝♦ ❞❡ r❛✐♦ κ✱ é ✐♠♣r♦✈á✈❡❧ ❛ ❞❡❝♦✲
❞✐✜❝❛çã♦ ❝♦rr❡t❛ ❞❡ r✳
❯♠ ❝ó❞✐❣♦ C s♦❜r❡ ✉♠ ❛❧❢❛❜❡t♦ A ♣♦ss✉✐ três ♣❛râ♠❡tr♦s ❢✉♥❞❛♠❡♥t❛✐s [n, M, d]✱
q✉❡ sã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦✱ ♦ s❡✉ ♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❡ ❛ s✉❛ ❞✐stâ♥❝✐❛ ♠í♥✐♠❛✳ ◆❡♠ s❡♠♣r❡ ❡①✐st❡ ✉♠ ❝ó❞✐❣♦ ❝♦♠ ♣❛râ♠❡tr♦s [n, M, d] ❞❡✜♥✐❞♦s
✐♥✐❝✐❛❧♠❡♥t❡✱ ♣♦✐s ❡①✐st❡ ✉♠❛ ✐♥t❡r❞❡♣❡♥❞ê♥❝✐❛ ❡♥tr❡ ❡ss❡s ♣❛râ♠❡tr♦s✳
✸✳✷ ■s♦♠❡tr✐❛s
❉❡✜♥✐çã♦ ✸✳✷✳✶✳ ❙❡❥❛ A ✉♠ ❛❧❢❛❜❡t♦ ❡ n ∈N✳ ❉✐③✲s❡ q✉❡ ✉♠❛ ❢✉♥çã♦ F :An−→An
é ✉♠❛ ✐s♦♠❡tr✐❛ s❡ ❡❧❛ ♣r❡s❡r✈❛ ❞✐stâ♥❝✐❛s ❞❡ ❍❛♠♠✐♥❣✱ ✐st♦ é✱
d(F(①), F(②)) =d(①,②) ∀①,②∈An.
➱ ❝❧❛r♦ q✉❡ t♦❞❛ ✐s♦♠❡tr✐❛ é ✉♠❛ ❜✐❥❡çã♦✱ ♣♦✐s✱ s❡❥❛ F ✉♠❛ ✐s♦♠❡tr✐❛✱ ❞❛❞♦s
①,②∈ An✱ s❡ F(①) = F(②)⇔ d(F(①), F(②)) = 0✱ ❝♦♠♦ d(F(①), F(②)) = d(①,②) = 0✱
s❡❣✉❡ q✉❡ ① = ②✱ ♠♦str❛♥❞♦ q✉❡ F é ✐♥❥❡t♦r❛❀ ❝♦♠♦ t♦❞❛ ❛♣❧✐❝❛çã♦ ✐♥❥❡t♦r❛ ❞❡ ✉♠
❝♦♥❥✉♥t♦ ✜♥✐t♦ ♥❡❧❡ ♣ró♣r✐♦ é s♦❜r❡❥❡t♦r❛✱ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳ Pr♦♣♦s✐çã♦ ✸✳✷✳✶✳ ➱ ✈á❧✐❞♦ ❛✜r♠❛r✿
✶✳ ❆ ❢✉♥çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ An é ✉♠❛ ✐s♦♠❡tr✐❛❀
✷✳ ❙❡ F é ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ An✱ ❡♥tã♦ F−1 é ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ An❀
✸✳ ❙❡ F ❡ G sã♦ ✐s♦♠❡tr✐❛s ❞❡ An✱ ❡♥tã♦ F ◦G é ✉♠❛ ✐s♦♠❡tr✐❛ ❞❡ An✳
❉❡♠♦♥str❛çã♦✿
1. ➱ ✐♠❡❞✐❛t❛✱ ♣♦✐s ❣❡r❛ ❛ ❞❡✜♥✐çã♦❀
2. ❈♦♠♦ F é ✉♠❛ ✐s♦♠❡tr✐❛✱ ❧♦❣♦ ❜✐❥❡t♦r❛✱ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ F−1 t❛❧ q✉❡
(F ◦F−1)(x) = (F−1 ◦F)(x) =x✳ ❉❛í✱
d(①,②) = d(F(F−1)(①), F(F−1)(②)) =d(F−1(①), F−1(②))
