Exercício 8 Considere o conjunto universo U=1,2,3,4,5,6,7,8 e os subconjuntos
X=1,2,3,4,5,5, Y=1,2,3 e Z=4,6,8
Determine:
  • X^C, Y^C e Z^C
  • X cap Y, X cap Z e Y cap Z
  • (X cap Y)^C e X^C cap Y^C
  • X^C cap X e X^C cup X
  • (X cup Y)^C e X^C cup Y^C
  • (X cap Z)^C e X^C cap Z^C

Passo 1

O enunciado dá o universo
U = 1,2,3,4,5,6,7,8
e os subconjuntos
X = 1,2,3,4,5, Y = 1,2,3, Z = 4,6,8
Precisamos usar as operações de teoria de conjuntos: complemento A^C=Usetminus A, interseção Acap B e união Acup B, além das Leis de De Morgan.

Passo 2

Calcular os complementos usando A^C=Usetminus A:
X^C = 6,7,8, Y^C = 4,5,6,7,8, Z^C = 1,2,3,5,7

Passo 3

Calcular as interseções:
Xcap Y = 1,2,3, Xcap Z = 4, Ycap Z = varnothing

Passo 4

Comparar (Xcap Y)^C e X^Ccap Y^C:
(Xcap Y)^C = Usetminus1,2,3 = 4,5,6,7,8,
X^Ccap Y^C = 6,7,8cap4,5,6,7,8 = 6,7,8

Passo 5

Calcular X^Ccap X e X^Ccup X:
X^Ccap X = varnothing,
X^Ccup X = U = 1,2,3,4,5,6,7,8

Passo 6

Comparar (Xcup Y)^C e X^Ccup Y^C:
Xcup Y = 1,2,3,4,5, (Xcup Y)^C = 6,7,8,
X^Ccup Y^C = 6,7,8cup4,5,6,7,8 = 4,5,6,7,8

Passo 7

Comparar (Xcap Z)^C e X^Ccap Z^C:
Xcap Z = 4, (Xcap Z)^C = Usetminus4 = 1,2,3,5,6,7,8,
X^Ccap Z^C = 6,7,8cap1,2,3,5,7 = 7

Resposta

  • X^C = 6,7,8,;Y^C = 4,5,6,7,8,;Z^C = 1,2,3,5,7
  • Xcap Y = 1,2,3,;Xcap Z = 4,;Ycap Z = varnothing
  • (Xcap Y)^C = 4,5,6,7,8,;X^Ccap Y^C = 6,7,8
  • X^Ccap X = varnothing,;X^Ccup X = 1,2,3,4,5,6,7,8
  • (Xcup Y)^C = 6,7,8,;X^Ccup Y^C = 4,5,6,7,8
  • (Xcap Z)^C = 1,2,3,5,6,7,8,;X^Ccap Z^C = 7

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