A, B dan C himpunan sembarang dan KC komplemen dari K. Maka A ∩ (B∪C) = ⋯
(1)  (A ∩ B) ∪ (A ∩ C)
(2)  (A ∪ B) ∩ (A ∪ C)
(3)  (AC ∪ (B ∪ C)C)C  
(4)  (AC ∪ (B ∪ C)C)C   

Jawab: (1) dan (3)

Sifat-sifat operasi himpunan:

  • Kesamaan
    • A ∪ A = A
    • A ∩ A = A
    • A – A = ∅
  • Komutatif
    • A ∪ B = B ∪ A
    • A ∩ B = B ∩ A
    • A + B = B + A
  • 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,

  • Berdasar sifat distributif:
    • A ∩ (B∪C) = (A ∩ B)∪(A ∩ C)
    • Pernyataan (1) benar

  • Berdasar Dalil De Morgan:  
    • A ∩ (B∪C) = ((A ∩ (B∪C))C)C 
    • A ∩ (B∪C) = (AC ∪(B ∪ C)C)C 
    • Pernyataan (3) benar


Sehingga untuk A, B dan C himpunan sembarang dan KC komplemen dari K. Maka A ∩ (B∪C) = (A ∩ B) ∪ (A ∩ C) dan A ∩ (B∪C) = (AC ∪ (B ∪ C)C)C.

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