Introdu\303\247\303\243o \303\240s Fun\303\247\303\265es: Defini\303\247\303\243o de Rela\303\247\303\243o Bin\303\241ria Como diz o ditado, voc\303\252 pode escolher seus amigos, mas voc\303\252 n\303\243o pode escolher seus parentes. Se n\303\263s escrevermos um par ordenado LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJBRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJCRidGNEY3RjpGN0ZBRjdGQUY3RkE= significa "pessoa A est\303\241 relacionada \303\240 pessoa B", ent\303\243o poder\303\255amos considerar cole\303\247\303\265es de pessoas que est\303\243o relacionadas entre si: 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 . A maior cole\303\247\303\243o poss\303\255vel incluem todos na Terra?

Nesta apresenta\303\247\303\243o, vamos definir os termos rela\303\247\303\243o bin\303\241ria, dom\303\255nio e imagem. Vamos supor que voc\303\252 esteja familiarizado com estes termos adicionais, que s\303\243o definidos em outras apresenta\303\247\303\265es desta cole\303\247\303\243o: gr\303\241fico e conjunto.

JSFH O que \303\251 uma rela\303\247\303\243o bin\303\241ria?Dado dois conjuntos de objetos, A e B, podemos considerar os pares LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJiRidGNEY3RjpGN0ZBRjdGQUY3RkE=, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn \303\251 um dos objectos em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn \303\251 um dos objetos em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn. Uma rela\303\247\303\243o bin\303\241ria sobre os conjuntos LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn e LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn \303\251 uma cole\303\247\303\243o de tais pares.O dom\303\255nio de uma rela\303\247\303\243o bin\303\241ria \303\251 o conjunto de todos os valores que aparecem como o primeiro componente na rela\303\247\303\243o. O imagem de uma rela\303\247\303\243o bin\303\241ria \303\251 o conjunto de todos os valores que aparecem como o segundo componente na rela\303\247\303\243o. Na matem\303\241tica, \303\251 comum referir-se aos componentes de pares ordenados de uma rela\303\247\303\243o bin\303\241ria como a entrada e sa\303\255da, respectivamente. Um exemplo de uma rela\303\247\303\243o bin\303\241ria \303\251 a posse de carro, onde o conjunto A s\303\243o todas as pessoas e o conjunto B s\303\243o todos os tipos de carros. Algumas pessoas n\303\243o tem nenhum carro, ent\303\243o elas n\303\243o aparecem na rela\303\247\303\243o em todas. Algumas pessoas tem um carro, ent\303\243o elas aparecem como a primeira componente de exatamente um par ordenado na rela\303\247\303\243o. Algumas pessoas possuem mais de um carro, e assim aparecem em mais de um par ordenado na rela\303\247\303\243o. O dom\303\255nio dessa rela\303\247\303\243o bin\303\241ria \303\251 o conjunto de todas as pessoas que possuem pelo menos um carro. A imagem \303\251 o conjunto de todos os tipos de carros de propriedade de pelo menos uma pessoa. JSFH

H\303\241 uma variedade de maneiras que uma rela\303\247\303\243o bin\303\241ria pode ser representada: como uma listagem de todos os pares na rela\303\247\303\243o, como um gr\303\241fico, como uma f\303\263rmula, e assim por diante.Suponha que temos um conjunto A de 4 n\303\272meros, 1, 2, 3 e 4, e um conjunto B de 3 letras, a, b, e c. Uma rela\303\247\303\243o entre A e B ser\303\241 uma cole\303\247\303\243o de pares, onde a primeira componente \303\251 um dos 4 n\303\272meros de A e a segunda componente \303\251 uma das 3 letras de B. No painel da esquerda do espa\303\247o de trabalho abaixo, voc\303\252 pode escolher exatamente quais pares est\303\243o na rela\303\247\303\243o. Os resultados de suas sele\303\247\303\265es ser\303\243o exibidos como uma lista de pares no painel esquerdo, e em duas representa\303\247\303\265es gr\303\241ficas nos pain\303\251is do meio e do direito. De A:,1,2,3,4M4_1_1:-CB_Select();M4_1_1:-B_Update();Para 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A resposta \303\240 quest\303\243o colocada no in\303\255cio desta folha de trabalho pode depender de sua compreens\303\243o de onde todos viemos. Entretanto, algumas quest\303\265es similares podem ter respostas mais claras. Por exemplo, considere a rela\303\247\303\243o R definida por: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJiRidGNEY3RjpGN0ZBRjdGQUY3RkE= est\303\241 em R, se a pessoa a \303\251 um ancestral da pessoa b (biol\303\263gico ou adotivo\342\200\224n\303\243o vamos ficar muito t\303\251cnico, aqui). O dom\303\255nio de R s\303\243o todas as pessoas que tiveram pelo menos uma descend\303\252ncia e a imagem de R s\303\243o todas as pessoas. Vamos dar uma olhada em um subconjunto espec\303\255fico da rela\303\247\303\243o R. Seja P o conjunto composto por seus pais. Para voc\303\252, quais s\303\243o todos os poss\303\255veis pares LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJwRicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJzRidGNEY3RjpGN0ZBRjdGQUY3RkE= em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn \303\251 um membro de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn? O conjunto de todas as pessoas que poderiam aparecer na segunda componente deste par (como s) \303\251 o conjunto de voc\303\252 e seus irm\303\243os e alguns de seus ou filhos, netos deles, etc. Qu\303\243o grande \303\251 este conjunto para voc\303\252?

No mundo da matem\303\241tica, considere a rela\303\247\303\243o raiz quadrada, S, onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYlLUYjNictSSNtaUdGJDYmUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSIsRidGNy9GO1Enbm9ybWFsRicvJSZmZW5jZUdGOS8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYmUSJ5RidGNEY3RjpGN0ZBRjdGQS1GPjYuUSJ+RidGN0ZBRkMvRkZGOUZHRklGS0ZNRk9GUS9GVUZTRjdGQQ==est\303\241 em S significa que y \303\251 uma raiz quadrada de LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JlEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn, ou, alternativamente, que 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. O dom\303\255nio dessa rela\303\247\303\243o \303\251 o conjunto de todos os n\303\272meros n\303\243o-negativos, e a imagem \303\251 o conjunto de todos os n\303\272meros, desde que todo n\303\272mero positivo tenha ambos uma raiz quadrada positiva e uma negativa. Podemos representar visualmente esta rela\303\247\303\243o em um gr\303\241fico, onde os valores de entrada (n\303\272meros n\303\243o-negativos) est\303\243o no eixo horizontal e os valores de sa\303\255da (ra\303\255zes quadradas positivas e negativas) est\303\243o no eixo vertical. 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