Jika himpunan P dan himpunan Q berpotongan sedangkan PC dan QC berturut-turut adalah komplemen dari P dan Q, maka (P ∩ Q) ∪ (P ∩ QC) = . . .
(A)   PC
(B)   QC
(C)   Q
(D)   P
(E)   PC ∩ QC 

Jawab: (D)

Sifat-sifat operasi himpunan:

  • Dalil De Morgan
    • (A ∩ B)C = AC ∪ BC
    • (A ∪ B)C = AC ∩ BC
  • Asosiatif:
    • A ∩ (B ∩ C) = (A ∩ B) ∩ C
    • A ∪ (B ∪ C) = (A ∪ B) ∪ C
  • Distributif:
    • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
    • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Sehingga,

(P ∩ Q) ∪ (P ∩ QC)

= (P ∪ (P ∩ QC)) ∩ (Q ∪ (P ∩ QC))

= (P ∪ (P ∩ QC)) ∩ (Q ∪ (QC ∩ P))

= (P ∩ (P ∪ QC) ∩ (S ∩ (Q ∪ P))

= P ∩ (Q ∪ P)

= P

Jadi, (P ∩ Q) ∪ (P ∩ QC) = P

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