Diagrama de Venn (Conjuntos)_ Exercícios Resolvidos

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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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Dicas de Concursos » Matemática
Ingrid Nunes em 08/01/2019 Comunicar erro!
John Venn criou, através de análises lógicas, um método de diagramação que consiste
simploriamente em uma relação entre círculos capaz de demonstrar operações entre conjuntos
Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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Agora devemos incluir os números cedidos aos seus lugares correspondentes no nos círculos.
Porém, devemos sempre começar pela intersecção entre todos os conjuntos sendo ela o valor
correspondente aos alunos que compraram tanto a pizza quanto o salgado. Veja:
Após isso, utilizaremos a informação dos dezesseis alunos que compraram salgado. Se inserirmos o
valor cedido no espaço correspondente aos salgados, teremos um diagrama errado. Veja o porquê:
Se 16 alunos compraram salgado, estamos considerando também os 4 alunos que além do salgado
compraram também a pizza.
31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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O espaço em laranja corresponde aos alunos que compraram apenas os alunos que compraram o
salgado, sem incluir aqueles que também compraram pizza. Justamente por isso não podemos incluir
o número 16 diretamente no espaço em laranja, mas sim subtrair 16 do número de alunos que
compraram os dois lanches para encontrar aqueles que compraram apenas o salgado.
n(A - B) = n(A) - n(A ∩ B)
16 – 4 = 12
Agora, a incógnita X representa a informação pedida pela questão, ou seja, os alunos que só
compraram pizza.
Se ao total temos 35 alunos, 7 nada compraram, 12 compraram apenas salgado e 4 pizza e salgado,
se relacionarmos esses elementos ao X em uma igualdade poderemos encontrar o valor da
incógnita.
35 = 12 + 4 + 7 + X 
35 = 23 + X 
X = 12
Ou seja, 12 alunos compraram apenas pizza.
2) Numa pesquisa, veri�cou-se que, das pessoas consultadas, 100 se informavam pelo site A; 150 por
meio do site B; 20 buscavam se informar por meio dos dois sites, A e B; e 110 não se informavam por
nenhum desses dois sites. Desse modo, é correto a�rmar que o número de pessoas consultadas
nessa pesquisa foi de?
Resolução:
Sabemos que o conjunto A possui 100 pessoas o B possui 150 e a intersecção entre A e B equivale a
20 pessoas. Podemos retirar do enunciado, também, que 110 pessoas fazem parte da pesquisa, mas
não fazem parte do conjunto A ou B, mas mesmo assim pertencem ao conjunto universo.
31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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Como ensinado anteriormente, sempre começamos pela intersecção entre todos os conjuntos, veja
abaixo:
Se 100 pessoas fazem parte do conjunto A e já temos 20 pessoas dentro do círculo correspondente a
A, nos restam 80 pessoas para completar o total. Agora, o conjunto B possui 150 pessoas no total,
entretanto devemos contabilizar as 20 pessoas que pertence à intersecção entre A e B, ou seja, as
que informaram A e B. Sendo assim, 150 – 20 = 130. Desse modo obtemos o número de pessoas que
informaram apenas o site B.
Se a questão pede o total de pessoas consultadas, precisamos apenas somar todos os elementos
presentes no diagrama.
80 + 20 + 130 + 110 = 340
Ou seja, foram entrevistadas 340 pessoas.
3) Uma sorveteria, com o intuito de otimizar suas vendas, fez uma pesquisa para saber qual era o
sorvete que seus clientes mais consomem. Entre os entrevistados, 130 consomem o sorvete de
morango; 165 consomem o sorvete de chocolate e 175 consomem o sorvete de �ocos. Dentro desse
grupo, 120 consomem apenas sorvete de �ocos; 25 consomem sorvete de morango e chocolate; 15
consomem o sorvete de morango e �ocos; 10 consomem sorvete de morango, �ocos e de chocolate.
Quantos clientes consomem os sorvetes de chocolate e �ocos?
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Resolução:
Começaremos pelos clientes que consomem os 3 sabores de sorvete, ou seja, a intersecção entre os
três conjuntos.
Sabemos que a questão pede o valor da intersecção entre os sorvetes de chocolate e �ocos, é preciso
colocar uma incógnita no espaço correspondente no diagrama de Venn. Além disso, podemos
preencher os espaços cujos valores foram informados no enunciado.
Se o total de consumidores do sorvete de �ocos é 175, podemos relacionar todos os elementos do
conjunto F com a incógnita através de uma igualdade resultará na resposta pedida pela questão.
175 = 15 + 10 + 120 + X 
175 = 145 + X 
X = 30
Ou seja, 30 clientes consomem o sorvete de chocolate e de �ocos.
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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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P U B L I C I D A D E
P U B L I C I D A D E
numéricos.
O Diagrama de Venn é um método que, através do desenho de �guras planas, os círculos, é
utilizado para resolver questões de conjuntos numéricos.
Por outro lado, existe também a fórmula (A U B) = n(A) + n(B) - n(A ∩ B) que nos permite realizar a
união de três elementos.
Através do diagrama, as operações dos conjuntos são representadas nos círculos e suas
interações. No diagrama abaixo, são representados três conjuntos.
Os espaços em azul representam as intersecções entre dois conjuntos de cada vez e o espaço
central em vermelho a intersecção entre todos os três conjuntos.
Representação de um único conjunto
Conjunto P dos números pares menores do que 12, P = {2, 4, 6, 8, 10}
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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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P U B L I C I D A D E
Representação de dois conjuntos
Conjuntos A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} e B = {1, 3, 5, 7, 9, 11, 13, 15}.
Nesses dois conjuntos temos que A U B (união) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15} e que A ∩ B
(intersecção) = {1, 3, 5, 7, 9}.
PESQUISAS PATROCINADAS
diagrama de venn conjuntos exercicios conjuntos exercicios resolvidos
exercicios enem para imprimir conjuntos chaves
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31/07/2020 Diagrama de Venn (Conjuntos): Exercícios Resolvidos
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Representação de três conjuntos
Dados os conjuntos A = {2, 4, 6, 8, 10, 12}, B = {3, 6, 9, 12} e C = {4, 8, 12, 16}.
Nesses conjuntos temos que A ∩ B = {6}, A ∩ C = {4, 8}, B ∩ C = {Ø} e A ∩ B ∩ C = {12}.
Exercícios resolvidos utilizando o diagrama de Venn
Veja os exemplos abaixo:
1) Em uma sala de aula, a professora de Matemática decidiu fazer um levantamento dos lanches
comprados pelos alunos. A professora veri�cou que, de um total de 35 alunos, dezesseis
compraram salgado; destes, quatro compraram pizza e salgado, e sete alunos não compraram
lanche nesse dia. Quantos alunos compraram apenas pizza?
Resolução:
Em um primeiro momento, precisamos identi�car que na questão existem dois conjuntos: o conjunto
dos alunos que compraram salgado e o conjunto dos alunos que compraram pizza.
Em segundo lugar, precisamos considerar que no universo dos 35 alunos existem 7 que não
compraram lanche e, sendo assim, não estarão inclusos nos círculos dos conjuntos.
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O Diagrama de Venn é um método que, através de o desenho de �guras planas, os círculos, é
utilizado para resolver questões de conjuntos numéricos. ...
Relação entre Conjuntos Numéricos - Igualdade, Inclusão/Continência e Pertinência
A teoria dos conjuntos é crucial para o entendimento de posteriores noções sobre o assunto
matemático dos conjuntos. Veja a Relação entre Conjuntos ...
Fórmulas Conjuntos Numéricos: Exercícios Resolvidos
Outra alternativa para a resolução de exercícios que envolvam conjuntos é através das fórmulas.
Embora o diagrama de Venn seja um método e�ciente e ...
Simulados Questões Provas Material
P U B L I C I D A D E
Por homet em 05/07/2020 às 12:46:14
Num levantamento entre 92 estudantes sobre o estudo de idiomas foram obtidos. 40
inglês, 28 francês, 26 espanhol, 13 inglês e francês, 8 estudam francês e espanhol, 20
inglês e espanhol, 4 estudam 3 idiomas. Quanto a estudantes estudam apenas um
desses idiomas?
Comentários em "Diagrama de Venn (Conjuntos): Exercícios Resolvidos"
https://www.gabarite.com.br/dica-concurso/235-conjunto-dos-numeros-naturais-inteiros-racionais-irracionais-e-complexos
https://www.gabarite.com.br/dica-concurso/248-operacoes-com-conjuntos-numericos-uniao-interseccao-e-diferenca
https://www.gabarite.com.br/dica-concurso/334-tipos-de-conjuntos-vazio-universo-unitario-finito-e-dos-numeros-infinitos
https://www.gabarite.com.br/dica-concurso/311-diagrama-de-venn-conjuntos-exercicios-resolvidos
https://www.gabarite.com.br/dica-concurso/246-relacao-entre-conjuntos-numericos-igualdade-inclusaocontinencia-e-pertinencia
https://www.gabarite.com.br/dica-concurso/294-formulas-conjuntos-numericos-exercicios-resolvidos
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