Resultado da API do MediaWiki.
Esta é a representação em HTML do formato JSON. O HTML é bom para o despiste de erros, mas inadequado para uso na aplicação.
Especifique o parâmetro format para alterar o formato de saída. Para ver a representação que não é em HTML do formato JSON, defina format=json.
Consulte a documentação completa, ou a ajuda da API para mais informação.
{
"compare": {
"fromid": 1,
"fromrevid": 1,
"fromns": 0,
"fromtitle": "P\u00e1gina principal",
"toid": 2,
"torevid": 2,
"tons": 0,
"totitle": "Portal:Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica/Conjuntos",
"*": "<tr><td colspan=\"2\" class=\"diff-lineno\" id=\"mw-diff-left-l1\">Linha 1:</td>\n<td colspan=\"2\" class=\"diff-lineno\">Linha 1:</td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\"><strong>O MediaWiki foi instalado</del>.<del class=\"diffchange diffchange-inline\"></strong></del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Em Matem\u00e1tica, '''conjunto''' \u00e9 uma cole\u00e7\u00e3o de objetos (chamados '''elementos'''). Os elementos podem representar qualquer coisa &mdash; n\u00fameros, pessoas, letras, etc - at\u00e9 mesmo outros conjuntos. Um conjunto pode conter outro(s) conjunto(s), inclusive. Incorretamente chamada de \"Teoria dos Conjuntos\" no ensino m\u00e9dio. Essa teoria existe, mas n\u00e3o \u00e9 tratada no ensino m\u00e9dio, sendo a Teoria mais conhecida, a Axiom\u00e1tica de Zermello Frankel (ZFC, C relacionado ao Axioma da Escolha), tratada de forma elementar no livro \"Teoria Ing\u00eanua dos Conjuntos\" de Paul Halmos, traduzida para o portugu\u00eas pelo prof. Irineu Bicudo</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">Consulte a [https://www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org/wiki/Special:MyLanguage/Help:Contents Ajuda do MediaWiki] para informa\u00e7\u00f5es sobre o uso do software wiki</del>.</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Trata-se de um ''conceito primitivo''</ins>. <ins class=\"diffchange diffchange-inline\">Um conjunto possui como \u00fanica propriedade os elementos que cont\u00e9m</ins>. <ins class=\"diffchange diffchange-inline\">Ou seja, dois conjuntos s\u00e3o iguais se eles tem os mesmos elementos</ins>.</div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>== <del class=\"diffchange diffchange-inline\">Onde come\u00e7ar </del>==</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>==<ins class=\"diffchange diffchange-inline\">Representa\u00e7\u00e3o</ins>==</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Matematicamente o conjunto \u00e9 representado por uma letra do alfabeto [[Latim|latino]], mai\u00fascula (A, B, C, ...). J\u00e1 os elementos do conjunto s\u00e3o representados por letras latinas min\u00fasculas. E a representa\u00e7\u00e3o completa do conjunto envolve a coloca\u00e7\u00e3o dos elementos entre chaves, da seguinte maneira:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{ v,x,y,z \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Para um conjunto ''A'' de 4 elementos ''v'', ''x'', ''y'' e ''z''</ins></div></td></tr>\n<tr><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-deleted\"><br></td><td class=\"diff-marker\"></td><td class=\"diff-context diff-side-added\"><br></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Configuration_settings Lista </del>de <del class=\"diffchange diffchange-inline\">op\u00e7\u00f5es </del>de <del class=\"diffchange diffchange-inline\">configura\u00e7\u00e3o</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A exce\u00e7\u00e3o \u00e9 feita a conjuntos que contenham elementos que devem ser representados por letras mai\u00fasculas &mdash; por exemplo, pontos geom\u00e9tricos:</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">FAQ Perguntas </del>e <del class=\"diffchange diffchange-inline\">respostas frequentes sobre </del>o <del class=\"diffchange diffchange-inline\">MediaWiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>S = \\{A, B, C, D \\}</math></ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* [<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">postorius</del>/<del class=\"diffchange diffchange-inline\">lists</del>/<del class=\"diffchange diffchange-inline\">mediawiki</del>-<del class=\"diffchange diffchange-inline\">announce</del>.<del class=\"diffchange diffchange-inline\">lists</del>.<del class=\"diffchange diffchange-inline\">wikimedia</del>.<del class=\"diffchange diffchange-inline\">org</del>/ <del class=\"diffchange diffchange-inline\">Subscreva </del>a <del class=\"diffchange diffchange-inline\">lista </del>de <del class=\"diffchange diffchange-inline\">divulga\u00e7\u00e3o </del>de <del class=\"diffchange diffchange-inline\">novas vers\u00f5es do MediaWiki]</del></div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div>* <del class=\"diffchange diffchange-inline\">[https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Localisation#Translation_resources Regionalize </del>o <del class=\"diffchange diffchange-inline\">MediaWiki </del>para a <del class=\"diffchange diffchange-inline\">sua l\u00edngua</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Especificando conjuntos===</ins></div></td></tr>\n<tr><td class=\"diff-marker\" data-marker=\"\u2212\"></td><td class=\"diff-deletedline diff-side-deleted\"><div><del class=\"diffchange diffchange-inline\">* </del>[<del class=\"diffchange diffchange-inline\">https</del>://<del class=\"diffchange diffchange-inline\">www</del>.<del class=\"diffchange diffchange-inline\">mediawiki</del>.<del class=\"diffchange diffchange-inline\">org</del>/<del class=\"diffchange diffchange-inline\">wiki</del>/<del class=\"diffchange diffchange-inline\">Special</del>:<del class=\"diffchange diffchange-inline\">MyLanguage</del>/<del class=\"diffchange diffchange-inline\">Manual</del>:<del class=\"diffchange diffchange-inline\">Combating_spam Aprenda </del>a <del class=\"diffchange diffchange-inline\">combater </del><<del class=\"diffchange diffchange-inline\">i</del>><del class=\"diffchange diffchange-inline\">spam</del></<del class=\"diffchange diffchange-inline\">i</del>> <del class=\"diffchange diffchange-inline\">na sua wiki</del>]</div></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A maneira mais simples de representar [[w:Equa\u00e7\u00f5es alg\u00e9bricas|algebricamente]] um conjunto \u00e9 atrav\u00e9s de uma lista de seus elementos entre chaves ('''{ }'''), conforme descrito nas se\u00e7\u00f5es anteriores:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>P = \\{ 6,28,496 \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Informalmente, usa-se o sinal ... quando a regra de forma\u00e7\u00e3o do conjunto \u00e9 \u00f3bvia a partir da enumera\u00e7\u00e3o de alguns elementos. Por exemplo, os conjuntos abaixo, o primeiro com um n\u00famero finito, e o segundo com um n\u00famero infinito de elementos:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>Z_{100} = \\{ 0, 1, 2, ..., 99 \\}\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>N = \\{ 0, 1, 2, 3, 4, 5, ... \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Conjuntos que s\u00e3o elementos de outros conjuntos s\u00e3o representados com chaves dentro de chaves:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>T = \\{ \\{1,6\\}, \\{5,8\\} \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por\u00e9m h\u00e1 nota\u00e7\u00f5es alternativas para representar os conjuntos, como a chamada nota\u00e7\u00e3o de composi\u00e7\u00e3o do conjunto, que utiliza uma condi\u00e7\u00e3o ''P'' para definir os elementos do conjunto:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{x|P(x)\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">''P'' \u00e9 uma fun\u00e7\u00e3o na vari\u00e1vel ''x'' que tem o dom\u00ednio igual ao conjunto ''A''. A vari\u00e1vel ''x'' pode estar limitada por outro conjunto, indicando-se a rela\u00e7\u00e3o de pertin\u00eancia adequada. Por exemplo:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{ x \\in \\mathbb{R} | x^2 - 6x = -8 \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto ''A'' ser\u00e1 formado, de acordo com o desenvolvimento da equa\u00e7\u00e3o dada, por 2 e 4 (\u00fanicos n\u00fameros inteiros que satisfazem a condi\u00e7\u00e3o ''P'', ou seja, que tornam verdadeira a equa\u00e7\u00e3o). Logo, <math>A = \\{ 2,4 \\}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Um cuidado deve ser tomado com a propriedade ''P(x)'', j\u00e1 que a forma\u00e7\u00e3o de conjuntos atrav\u00e9s deste m\u00e9todo pode gerar resultados paradoxais.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Terminologia==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Conjunto unit\u00e1rio===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Um conjunto unit\u00e1rio possui um \u00fanico elemento.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Conjunto vazio===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Todo conjunto tamb\u00e9m possui como subconjunto '''o conjunto vazio''' representado por <math>\\{\\}\\!\\,</math>, <math>\\empty</math>, <math>\\varnothing</math> ou <math>\\phi\\!\\,</math>. Podemos mostrar isto supondo que se o conjunto vazio n\u00e3o est\u00e1 contido no conjunto em quest\u00e3o, ent\u00e3o o conjunto vazio deve possuir um elemento ao menos que n\u00e3o perten\u00e7a a este conjunto. Como o conjunto vazio n\u00e3o possui elementos, isto n\u00e3o \u00e9 poss\u00edvel. Como todos os conjuntos vazios s\u00e3o iguais uns aos outros, \u00e9 permiss\u00edvel falar de um \u00fanico conjunto sem elementos.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Conjuntos num\u00e9ricos===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Existem tamb\u00e9m os conjuntos num\u00e9ricos, que em considera\u00e7\u00e3o especial em matem\u00e1tica. Os principais conjuntos n\u00famericos s\u00e3o listados a seguir.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros naturais====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Os [</ins>[<ins class=\"diffchange diffchange-inline\">Portal</ins>:<ins class=\"diffchange diffchange-inline\">Forma\u00e7\u00e3o Intermedi\u00e1ria</ins>/<ins class=\"diffchange diffchange-inline\">Matem\u00e1tica</ins>/<ins class=\"diffchange diffchange-inline\">N\u00fameros naturais|'''n\u00fameros naturais''']] s\u00e3o usados para contar</ins>. <ins class=\"diffchange diffchange-inline\"> O s\u00edmbolo <math>\\mathbb{N}</math> usualmente representa este conjunto</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CaixaMsg|tipo=exemplo|texto= A li\u00e7\u00e3o sobre n\u00fameros naturais oferece informa\u00e7\u00f5es mais detalhadas sobre o assunto : '''[[Portal:Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica/N\u00fameros naturais|Li\u00e7\u00e3o N\u00fameros Naturais]]'''}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros inteiros====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos [[Portal:Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica/N\u00fameros inteiros|'''n\u00fameros inteiros''']] aparecem como solu\u00e7\u00f5es de equa\u00e7\u00f5es como <var>x</var> + <var>a</var> = <var>b<</ins>/<ins class=\"diffchange diffchange-inline\">var>. O s\u00edmbolo <math>\\mathbb{Z}<</ins>/<ins class=\"diffchange diffchange-inline\">math> usualmente representa este conjunto (do termo alem\u00e3o ''Zahlen'' que significa ''n\u00fameros'').</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">{{CaixaMsg|tipo=exemplo|texto= A li\u00e7\u00e3o sobre n\u00fameros inteiros oferece informa\u00e7\u00f5es mais detalhadas sobre o assunto : '''[[Portal</ins>:<ins class=\"diffchange diffchange-inline\">Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica</ins>/<ins class=\"diffchange diffchange-inline\">N\u00fameros inteiros|Li\u00e7\u00e3o N\u00fameros Inteiros]]'''}}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros racionais====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos [[Portal</ins>:<ins class=\"diffchange diffchange-inline\">Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica/N\u00fameros racionais|'''n\u00fameros racionais''']] s\u00e3o todos os n\u00fameros que podem ser representados por fra\u00e7\u00f5es (e s\u00e3o expressos tanto na forma fracion\u00e1ria quanto na forma decimal - por exemplo 3/4 e 0,75). Eles aparecem como solu\u00e7\u00f5es de equa\u00e7\u00f5es como <var>a</var> + <var>bx</var> = <var>c</var>. O s\u00edmbolo <math>\\mathbb{Q}</math> usualmente representa este conjunto (da palavra quociente).</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''T\u00f3picos'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros racionais#N\u00fameros racionais e fra\u00e7\u00f5es|N\u00fameros racionais e fra\u00e7\u00f5es]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros racionais#Defini\u00e7\u00f5es|Defini\u00e7\u00f5es]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros racionais#Decimais|Decimais]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros racionais#Tipos </ins>de <ins class=\"diffchange diffchange-inline\">fra\u00e7\u00f5es|Tipos </ins>de <ins class=\"diffchange diffchange-inline\">fra\u00e7\u00f5es]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>*[<ins class=\"diffchange diffchange-inline\">[Matem\u00e1tica elementar/Conjuntos/N\u00fameros racionais#Opera\u00e7\u00f5es|Opera\u00e7\u00f5es]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros irracionais====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos [[Portal</ins>:<ins class=\"diffchange diffchange-inline\">Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica</ins>/<ins class=\"diffchange diffchange-inline\">N\u00fameros irracionais|'''n\u00fameros irracionais''']] cont\u00e9m todos os n\u00fameros que n\u00e3o podem ser representados por fra\u00e7\u00f5es do tipo '''p'''</ins>/<ins class=\"diffchange diffchange-inline\">'''q''', onde '''p''' e '''q''' s\u00e3o [[Portal:Forma\u00e7\u00e3o Intermedi\u00e1ria/Matem\u00e1tica/N\u00fameros inteiros|'''n\u00fameros inteiros''']], com '''q''' diferente de zero</ins>. <ins class=\"diffchange diffchange-inline\">Estes n\u00fameros podem, no entanto, ser associados a pontos numa reta, a reta real</ins>. <ins class=\"diffchange diffchange-inline\">O s\u00edmbolo <math>\\mathbb{I}</math> usualmente representa este conjunto.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros reais====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos [[Matem\u00e1tica elementar</ins>/<ins class=\"diffchange diffchange-inline\">Conjuntos</ins>/<ins class=\"diffchange diffchange-inline\">N\u00fameros reais|'''n\u00fameros reais''']] \u00e9 uma expans\u00e3o do conjunto dos n\u00fameros racionais que engloba n\u00e3o s\u00f3 os inteiros e os fracion\u00e1rios, positivos e negativos, mas tamb\u00e9m todos os n\u00fameros irracionais. Os n\u00fameros reais podem ser dispostos ordenadamente em uma reta que \u00e9 chamada reta real.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Imagem</ins>:<ins class=\"diffchange diffchange-inline\">Real Number Line.svg|Reta num\u00e9rica real]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''T\u00f3picos'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Potencia\u00e7\u00e3o|Potencia\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Defini\u00e7\u00e3o|Defini\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Propriedades da potencia\u00e7\u00e3o|Propriedades da potencia\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Radicia\u00e7\u00e3o|Radicia\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Propriedades da radicia\u00e7\u00e3o|Propriedades da radicia\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Racionaliza\u00e7\u00e3o de denominadores|Racionaliza\u00e7\u00e3o de denominadores]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Intervalos reais|Intervalos reais]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros reais#Exerc\u00edcios|Exerc\u00edcios]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros complexos====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos [[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos|'''n\u00fameros complexos''']] inclue os n\u00fameros, que resultam de qualquer radicia\u00e7\u00e3o poss\u00edvel, tendo uma parte imagin\u00e1ria e uma parte real.\u00a0 O s\u00edmbolo <math>\\mathbb{C}<</ins>/<ins class=\"diffchange diffchange-inline\">math> usualmente representa este conjunto. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Cada numero complexo \u00e9 a soma dos n\u00fameros reais e dos imagin\u00e1rios</ins>: <ins class=\"diffchange diffchange-inline\"><math>r + s\\imath</math>.\u00a0 Aqui tanto <var>r</var> quanto <var>s</var> podem ser iguais a zero; ent\u00e3o os conjuntos dos n\u00fameros reais </ins>e o <ins class=\"diffchange diffchange-inline\">dos imagin\u00e1rios s\u00e3o subconjuntos do conjunto dos n\u00fameros complexos.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''T\u00f3picos'''</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Introdu\u00e7\u00e3o|Introdu\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#O n\u00famero im\u00e1ginario|O n\u00famero im\u00e1ginario]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Formas de representar os complexos|Formas de representar os complexos]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Opera\u00e7\u00f5es com os complexos|Opera\u00e7\u00f5es com os complexos]</ins>]</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*</ins>*[<ins class=\"diffchange diffchange-inline\">[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Soma e subtra\u00e7\u00e3o|Soma e subtra\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Multiplica\u00e7\u00e3o|Multiplica\u00e7\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">**[[Matem\u00e1tica elementar/Conjuntos/N\u00fameros complexos#Divis\u00e3o|Divis\u00e3o]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto dos n\u00fameros imagin\u00e1rios====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O conjunto dos '''n\u00fameros imagin\u00e1rios puros''' inclui os n\u00fameros que aparecem como solu\u00e7\u00f5es de equa\u00e7\u00f5es como <var>x</var> <sup>2</sup> + r = 0 onde r > 0.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Outros conjuntos num\u00e9ricos====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">H\u00e1 outros conjuntos num\u00e9ricos definidos na matem\u00e1tica, mas que n\u00e3o interessam nesse n\u00edvel de estudo.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Exemplo:''' O conjunto dos [[w:N\u00famero alg\u00e9brico|'''n\u00fameros alg\u00e9bricos''']] inclue os n\u00fameros, que aparecem como solu\u00e7\u00f5es de [[w:Equa\u00e7\u00e3o Polinomial|equa\u00e7\u00f5es polinomiais]] (com coeficientes inteiros) e envolvem ra\u00edzes e alguns outros n\u00fameros irracionais. O s\u00edmbolo <math>\\mathbb{A}</math> ou <math>\\bar{\\mathbb{Q}}</math> usualmente representa este conjunto.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Subconjuntos===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dizemos que um conjunto ''A'' \u00e9 '''subconjunto''' de outro conjunto ''B'' quando todos os elementos de A tamb\u00e9m pertencem a B. Por exemplo:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::A = { 1,2,3 }</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::B = { 1,2,3,4,5,6 }</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nesse caso ''A'' \u00e9 subconjunto de ''B'', \u00e9 indica-se <math>A \\subset B</math>. Deve-se reparar que ''B'' \u00e9 subconjunto de si mesmo; os subconjuntos de ''B'' que n\u00e3o s\u00e3o iguais a ''B'' s\u00e3o chamados '''subconjuntos pr\u00f3prios'''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Nota:''' O conjunto vazio, { } ou \u0424 (phi), \u00e9 um subconjunto de todos os conjuntos.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">====Conjunto das partes ou pot\u00eancia====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dado um conjunto ''A'', definimos o '''conjunto das partes''' de ''A'', <math>\\mathcal{P}(A)</math>, como o conjunto que cont\u00e9m todos os subconjuntos de ''A'' (incluindo o ''conjunto vazio'' e o pr\u00f3prio conjunto ''A'').</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Uma maneira pr\u00e1tica de determinar <math>\\mathcal{P}(A)</math> \u00e9 pensar em todos os subconjuntos com um elemento, depois todos os subconjuntos com dois elementos, e assim por diante.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Exemplo''':</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se A = { 1, 2, 3 }, ent\u00e3o <math>\\mathcal{P}(A)</math> = { &empty;, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Observa\u00e7\u00e3o''': </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">Se o conjunto ''A'' tem ''n'' elementos, o conjunto <math>\\mathcal{P}(A)<</ins>/<ins class=\"diffchange diffchange-inline\">math> ter\u00e1 2<sup>''n''</sup> elementos. Ou seja:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>\\#\\mathcal{P}(A) = 2^{\\#A}<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Nota:''' O conjunto das partes \u00e9 uma [[w:\u00e1lgebra booleana|\u00e1lgebra booleana]] sobre as opera\u00e7\u00f5es de uni\u00e3o e interse\u00e7\u00e3o. </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O [[w:Teorema de Cantor|Teorema de Cantor]] estabelece que <math>|A| < |P(A)|</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Conjunto Universo===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Em certos problemas da teoria dos conjuntos, \u00e9 preciso que se defina um conjunto que contenha todos os conjuntos considerados. Assim, todos os conjuntos trabalhados no problema seriam subconjuntos de um conjunto maior, que \u00e9 conhecido como '''conjunto universo''', ou simplesmente ''universo''</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por exemplo: em um problema envolvendo conjuntos de n\u00fameros inteiros, o conjunto dos n\u00fameros inteiros '''Z''' \u00e9 o conjunto universo; em um problema envolvendo palavras (consideradas como conjuntos de letras), o universo \u00e9 o alfabeto</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Rela\u00e7\u00f5es entre conjuntos==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Rela\u00e7\u00e3o de inclus\u00e3o===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Rela\u00e7\u00e3o de pertin\u00eancia===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Se <math>\\,\\! a<</ins>/<ins class=\"diffchange diffchange-inline\">math> \u00e9 um elemento de <math>A \\,\\!<</ins>/<ins class=\"diffchange diffchange-inline\">math>, n\u00f3s podemos dizer que o elemento <math>a \\,\\!<</ins>/<ins class=\"diffchange diffchange-inline\">math> pertence ao conjunto <math>A \\,\\!</math> e podemos escrever <math>a \\in A</math>. Se <math>a \\,\\!</math> '''n\u00e3o''' \u00e9 um elemento de <math>A \\,\\!</math>, n\u00f3s podemos dizer que o elemento <math>a \\,\\!</math> '''n\u00e3o''' pertence ao conjunto <math>A \\,\\!</math> e podemos escrever <math>a \\not\\in A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Exemplos:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* <math></ins>-<ins class=\"diffchange diffchange-inline\">16 \\in \\mathbb{Z}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* <math>c \\in \\{a,b,c,d,e,f,g,h,i,j,k,l\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* <math>c\\ \\not\\in \\ \\{a,e,i,o,u\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* <math>\\frac{4}{9}\\ \\not\\in \\ \\mathbb{Z}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==== Subconjuntos pr\u00f3prios e impr\u00f3prios ====</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Se <math>A \\,\\!</math> e <math>B \\,\\!</math> s\u00e3o conjuntos e todo o elemento <math>x \\,\\!</math> pertencente a <math>A \\,\\!</math> tamb\u00e9m pertence a <math>B \\,\\!</math>, ent\u00e3o o conjunto <math>A \\,\\!</math> \u00e9 dito um [[#Subconjuntos|subconjunto]] do conjunto <math>B \\,\\!</math>, denotado por <math>A \\subseteq B</math>. Note que esta defini\u00e7\u00e3o inclui o caso em que <math>A</math> e <math>B</math> possuem os mesmos elementos, isto \u00e9, s\u00e3o o mesmo conjunto (<math>A=B</math>). Se <math>A \\subseteq B</math> e ao menos um elemento pertencente a <math>B \\,\\!</math> n\u00e3o pertence a <math>A \\,\\!</math>, ent\u00e3o <math>A \\,\\!</math> \u00e9 chamado de '''subconjunto pr\u00f3prio''' de <math>B \\,\\!</math>, denotado por <math>A \\subset B</math>. Todo conjunto \u00e9 subconjunto dele pr\u00f3prio, chamado de ''subconjunto impr\u00f3prio''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Igualdade de conjuntos===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dois conjuntos ''A'' e ''B'' s\u00e3o ditos '''iguais''' se, e somente se, t\u00eam os mesmos elementos</ins>. <ins class=\"diffchange diffchange-inline\">Ou seja, todo elemento de ''A'' \u00e9 elemento de ''B'' e vice-versa</ins>. <ins class=\"diffchange diffchange-inline\">A simbologia usada \u00e9 <math>A = B \\!\\,</math></ins>. <ins class=\"diffchange diffchange-inline\">Se um conjunto n\u00e3o \u00e9 igual a outro, utiliza-se o s\u00edmbolo <math>\\ne<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Simetria de conjuntos===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Um conjunto ''A'' \u00e9 dito sim\u00e9trico se, para todo elemento ''</ins>a<ins class=\"diffchange diffchange-inline\">'' pertencente a ele, houver tamb\u00e9m um elemento ''-a'' pertencente a esse conjunto. Os conjuntos num\u00e9ricos '''Z''', '''R''', '''Q''' e '''C''' s\u00e3o sim\u00e9tricos.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Opera\u00e7\u00f5es com conjuntos ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Uni\u00e3o===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A uni\u00e3o de dois conjuntos A e B \u00e9 um conjunto que cont\u00e9m todos os elementos </ins>de <ins class=\"diffchange diffchange-inline\">A, todos os elementos </ins>de <ins class=\"diffchange diffchange-inline\">B, e nada mais al\u00e9m disso. Ou ent\u00e3o: '''Dado um universo ''U'' e dois conjuntos ''A'' e ''B'', chama-se ''uni\u00e3o de A com B'' ao conjunto cujos elementos pertencem pelo menos ao conjunto ''A'' ou ao conjunto ''B''.''' Matematicamente:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A \\cup B = \\{ x \\in U | x \\in A \\lor x \\in B \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por exemplo:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{a,e,i\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>B = \\{o,u\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A \\cup B = \\{a,e,i,o,u\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{2,3,4,5\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>B = \\{1,3,5\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A \\cup B = \\{1,2,3,4,5\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Observar no \u00faltimo exemplo que os elementos repetidos (3,5) n\u00e3o aparecem na uni\u00e3o.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* A uni\u00e3o de um conjunto <math>A\\!\\,</math>, qualquer que seja, com o conjunto vazio \u00e9 igual ao pr\u00f3prio conjunto <math>A\\!\\,</math>, <math>A \\cup \\{\\} = A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>* <ins class=\"diffchange diffchange-inline\">Tamb\u00e9m deve ser observado que a opera\u00e7\u00e3o de uni\u00e3o \u00e9 comutativa, ou seja, <math>A \\cup (B \\cup C) = (A \\cup B) \\cup C = (A \\cup C) \\cup B</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Intersec\u00e7\u00e3o===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''A intersec\u00e7\u00e3o de dois conjuntos''' <math>A\\!\\,</math> e <math>B\\!\\,</math>, \u00e9 o conjunto de elementos que pertencem aos dois conjuntos. Ou ent\u00e3o: '''Dados dois conjuntos <math>A\\!\\,</math> e <math>B\\!\\,</math>, pertencentes a um universo ''U'', chama-se ''intersec\u00e7\u00e3o de A com B'' ao conjunto cujos elementos pertencem tanto a <math>A\\!\\,</math> quanto a <math>B\\!\\,</math>. Matematicamente:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A \\cap B = \\{ x \\in U | x \\in A \\land x \\in B \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por exemplo:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A = \\{1,2,3\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>B = \\{3,4,5\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A \\cap B = \\{3\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>C = \\{a,e,i,o,u,y\\}\\!\\,</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:</ins>:<ins class=\"diffchange diffchange-inline\"><math>D = \\{b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,z\\}\\!\\,<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>C \\cap D = \\{\\}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Observar no \u00faltimo exemplo que, dado os conjuntos n\u00e3o terem elementos iguais, a intersec\u00e7\u00e3o resulta num conjunto vazio</ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Diferen\u00e7a===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Dado um universo ''U'' ao qual pertencem dois conjuntos ''A'' e ''B'', chama-se ''diferen\u00e7a de A menos B'' ao conjunto de elementos que pertencem a ''A'' e n\u00e3o pertencem a ''B''; chama-se de ''diferen\u00e7a de B menos A'' ao conjunto de elementos que pertencem a ''B'' e n\u00e3o pertencem a ''A'''''</ins>. <ins class=\"diffchange diffchange-inline\">Matematicamente:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>A - B = \\{x \\in U | x \\in A \\land x \\not\\in B\\}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>B - A = \\{x \\in U | x \\in B \\land x \\not\\in A\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por exemplo, o conjunto definido pela diferen\u00e7a entre os n\u00fameros inteiros e n\u00fameros naturais \u00e9 igual ao conjunto Z<sub>-<</ins>/<ins class=\"diffchange diffchange-inline\">sub> (n\u00fameros inteiros n\u00e3o-positivos):</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:</ins>:<ins class=\"diffchange diffchange-inline\">'''Z''' = {...,-2,-1,0,1,2,...}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::'''N''' = {1,2,3,4,5,...}</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>\\mathbb{Z} - \\mathbb{N} = \\mathbb{Z}_{-} = \\{...,-2,-1,0\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* A subtra\u00e7\u00e3o de um conjunto ''A'' menos um conjunto vazio \u00e9 igual ao pr\u00f3prio conjunto ''A'', <math>A - \\{ \\} = A<</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Complementar===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Dado um universo ''U'', diz-se complementar de um conjunto ''A'', em rela\u00e7\u00e3o ao universo ''U'', </ins>o <ins class=\"diffchange diffchange-inline\">conjunto que cont\u00e9m todos os elementos presentes no universo e que n\u00e3o perten\u00e7am a ''A''.''' Tamb\u00e9m define-se complementar </ins>para <ins class=\"diffchange diffchange-inline\">dois conjuntos, contanto que um deles seja subconjunto do outro. Nesse caso, diz-se, por exemplo, ''complementar de B em rela\u00e7\u00e3o a A'' (sendo ''B'' um subconjunto de ''A'') &mdash; \u00e9 o complementar relativo &mdash; e usa-se o s\u00edmbolo <math>\\complement</math>. Matematicamente:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>\\complement B_A = \\{ x \\in A | x \\not\\in B \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Exemplo:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::A = { 3,4,9,{10,12},{25,27} }</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::D = { {10,12} }</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:::<math>\\complement D_A = \\{ 3,4,9,\\{25,27\\} \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Cardinalidade==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A '''cardinalidade''' de um conjunto ''A'' representa </ins>a <ins class=\"diffchange diffchange-inline\">quantidade de elementos do conjunto, e \u00e9 </ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">'''Exemplos''':</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se ''A'' = { 7, 8, 9 }, ent\u00e3o ''A'' = 3</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">:Se ''A'' = { }, ent\u00e3o ''A'' = 0.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Se um conjunto tem ''n'' elementos, onde ''n'' \u00e9 um [[Matem\u00e1tica elementar/Conjuntos/N\u00fameros naturais|n\u00famero natural]</ins>] <ins class=\"diffchange diffchange-inline\">(possivelmente 0), ent\u00e3o diz-se que o conjunto \u00e9 um '''conjunto finito''' com uma '''cardinalidade de n''' ou n\u00famero '''N\u00famero cardinal''' n.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Mesmo se o conjunto n\u00e3o possui um n\u00famero finito de elementos, pode-se definir a cardinalidade, gra\u00e7as ao trabalho desenvolvido pelo matem\u00e1tico [</ins>[<ins class=\"diffchange diffchange-inline\">w</ins>:<ins class=\"diffchange diffchange-inline\">Georg Cantor|Georg Cantor]]. Neste caso, a cardinalidade poder\u00e1 ser <math>\\aleph_0<</ins>/<ins class=\"diffchange diffchange-inline\">math> ([[w:aleph|aleph]] zero), <math>\\aleph_1, \\aleph_2 ...<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins>.</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Nos dois casos a cardinalidade de um conjunto <math>A</math> \u00e9 denotada por <math>|A|</math> ou por <math>\\#A</math></ins>. <ins class=\"diffchange diffchange-inline\">Se para dois conjuntos ''A'' e ''B'' \u00e9 poss\u00edvel fazer uma rela\u00e7\u00e3o um-a-um (ou seja, uma bije\u00e7\u00e3o) entre seus elementos, ent\u00e3o <math> |A|=|B| <</ins>/<ins class=\"diffchange diffchange-inline\">math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">== Produto cartesiano ==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Dados dois conjuntos ''A'' e ''B'', chama-se '''produto cartesiano de A em B''' ao conjunto formado por todos os pares ordenados cuja primeira coordenada seja pertencente a ''A'', e a segunda coordenada seja pertencente a ''B''. O simbolo do produto cartesioano \u00e9 <math>\\times<</ins>/<ins class=\"diffchange diffchange-inline\">math>. Matematicamente:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>:<ins class=\"diffchange diffchange-inline\">:<math>A \\times B = \\{(x,y) | x \\in A \\land y \\in B \\}<</ins>/<ins class=\"diffchange diffchange-inline\">math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">O [[w:Produto cartesiano|produto cartesiano]] de dois conjuntos ''A'' e ''B'' \u00e9 o conjunto de [[w:Par ordenado|pares ordenados]]:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: <math>A \\times B= \\{(a,b) </ins>: a <ins class=\"diffchange diffchange-inline\">\\in A \\land b \\in B\\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">A soma ou uni\u00e3o disjunta de dois conjuntos ''A'' e ''B'' \u00e9 o conjunto</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">: </ins><<ins class=\"diffchange diffchange-inline\">math>A + B = A \\times \\{0\\} \\cup B \\times \\{1\\}</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* O produto cartesiano \u00e9 n\u00e3o-comutativo: <math>A \\times B \\ne B \\times A</math>.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* Quem desenvolveu o conceito de produto cartesiano foi o matem\u00e1tico [[w:Ren\u00e9 Descartes|Descartes]], quando desenvolvia a geometria anal\u00edtica. Ele enunciou, por exemplo, que o produto cartesiano definido por dois conjuntos de n\u00fameros reais '''R''' (imagine os eixos das abcissas e ordenadas num gr\u00e1fico) \u00e9 igual a um plano.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">===Par ordenado===</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Um '''par ordenado''' \u00e9 uma cole\u00e7\u00e3o de dois objetos que tem uma ordem definida; existe o ''primeiro elemento'' (ou '''primeira coordenada''') e o ''segundo elemento'' (ou '''segunda coordenada'''). Diferentemente do conjunto { a,b }, um par ordenado &mdash; simbolizado por (a,b) &mdash; precisa ser apresentado em uma determinada ordem, e dois pares ordenados s\u00f3 s\u00e3o iguais quando os ''primeiros elementos'' s\u00e3o iguais e os ''segundos elementos'' s\u00e3o iguais. Ou seja,</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>(a,b) \\ne (b,a)</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Por\u00e9m, o par ordenado pode ser representado como um conjunto, tal que n\u00e3o existe ambiguidade quanto \u00e0 ordem. Esse conjunto \u00e9:</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math>(a,b) = \\{ \\{a\\}, \\{a,b\\} \\}</math></ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">::<math</ins>><ins class=\"diffchange diffchange-inline\">(b,a) = \\{ \\{b\\}, \\{b,a\\} \\}</ins></<ins class=\"diffchange diffchange-inline\">math</ins>></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">Observar que o formato do conjunto, que inclui um subconjunto contendo os dois elementos do par e um conjunto contendo o primeiro elemento, elimina a possibilidade de ambiguidade quanto \u00e0 ordem. A nota\u00e7\u00e3o (a,b) tamb\u00e9m \u00e9 conhecida como '''intervalo aberto'''.</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Refer\u00eancias==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[w:Conjunto|Conjunto]], artigo da [[w:Wikip\u00e9dia|Wikip\u00e9dia]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[w:Complementar|Complementar]], artigo da [[w:Wikip\u00e9dia|Wikip\u00e9dia]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[w:Diagrama de Venn|Diagrama de Venn]], artigo da [[w:Wikip\u00e9dia|Wikip\u00e9dia]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[w:Diagrama de Euler|Diagrama de Euler]], artigo da [[w:Wikip\u00e9dia|Wikip\u00e9dia]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[b:Matem\u00e1tica elementar/Conjuntos|Conjuntos]], no [[w:Wikilivros|Wikilivros]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">*[[w:Teoria dos conjuntos|Teoria dos conjuntos]], artigo da [[w:Wikip\u00e9dia|Wikip\u00e9dia]]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">==Liga\u00e7\u00f5es externas==</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/naturais/naturais1.htm N\u00fameros Naturais: Primeira parte]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/naturais/naturais2.htm N\u00fameros Naturais: Segunda parte]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/naturais/divisibilidade.htm Crit\u00e9rios de Divisibilidade]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/naturais/naturais2-a.htm Exerc\u00edcios Resolvidos de MDC, MMC e Divisores]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/inteiros/inteiros.htm N\u00fameros Inteiros]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/fracoes/fracoes.htm Fra\u00e7\u00f5es]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/fracoes/fracdec.htm Fra\u00e7\u00f5es e N\u00fameros Decimais]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/fracoes/racionais.htm N\u00fameros Racionais]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">* [http://pessoal.sercomtel.com.br/matematica/fundam/fracoes/fracoes-a.htm Fra\u00e7\u00f5es e N\u00fameros decimais (Exerc\u00edcios)]</ins></div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div>\u00a0</div></td></tr>\n<tr><td colspan=\"2\" class=\"diff-side-deleted\"></td><td class=\"diff-marker\" data-marker=\"+\"></td><td class=\"diff-addedline diff-side-added\"><div><ins class=\"diffchange diffchange-inline\">[[Categoria:Forma\u00e7\u00e3o Intermedi\u00e1ria-Matem\u00e1tica|Forma\u00e7\u00e3o Intermedi\u00e1ria-Matem\u00e1tica]</ins>]</div></td></tr>\n"
}
}