n(A) menyatakan banyaknya anggota himpunan A. Jika n(A – B) = 3x + 60, n(A ∩ B) = x, n(B – A) = 5x2, dan n(A ∪ B) = 300 maka n(A) = ….
(A)  100
(B)  150
(C)  240
(D)  250
(E)  275

Jawab: (C)

  • Dari soal diketahui:
    • n(A – B) = 3x + 60
    • n(A ∩ B) = x2
    • n(B – A) = 5x
    • n(A ∪ B) = 300
  • Pada operasi himpunan berlaku persamaan: n(S) = n(A ∪ B) + n(A ∪ B)C
    • Untuk n(A ∪ B)C = 0:
      n(S) = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
      
    • Untuk n(A ∪ B)C ≠ 0:
      n(S) – n(A∪B)C = n(A) + n(B) – n(A∩B)

Sehingga,

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)

n(A ∪ B) = n(A – B) + n(B – A) + n(A ∩ B)

300 = 3x + 60 + 5x + x2

x2 + 8x – 240 = 0

(x + 20)(x – 12) = 0

x1 = –20 atau x2 = 12

Diperoleh dua nilai x, namun karena banyaknya anggota himpunan tidak mungkin bernilai negatif maka nilai x yang memenuhi adalah nilai x yang positif yaitu x = 12.

Menghitung n(A):

n(A) = n(A – B) + n(A ∩ B)

n(A) = 3x + 60 + x2

n(A) = 3(12) + 60 + 122

n(A) = 36 + 60 + 144 = 240

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