Jawab:
Baik sifat distributif irisan terhadap gabungan, maupun sifat distributif gabungan terhadap irisan, berlaku terhadap himpunan A, B, dan C.

Penjelasan:

Diketahui: A = {a, b, c}. B = {b, c, d}, C = {c, d, e}

a. Sifat distributif irisan terhadap gabungan

• A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  ⇒ {a, b, c} ∩ {b, c, d, e} = {b, c} ∪ {c}
  ⇒ {b, c} = {b, c}
  ⇒ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) berlaku.
• B ∩ (A ∪ C) = (B ∩ A) ∪ (B ∩ C)
  ⇒ {b, c, d} ∩ {a, b, c, d, e} = {b, c} ∪ {c, d}
  ⇒ {b, c, d} = {b, c, d}
  ⇒ B ∩ (A ∪ C) = (B ∩ A) ∪ (B ∩ C) berlaku.
• C ∩ (A U B) = (C ∩ A) ∪ (C ∩ B)
  ⇒ {c, d, e} ∩ {a, b, c, d} = {c} ∪ {c, d}
  ⇒ {c, d} = {c, d}
  ⇒ C ∩ (A U B) = (C ∩ A) ∪ (C ∩ B) berlaku.

KESIMPULAN

∴  Sifat distributif irisan terhadap gabungan berlaku pada himpunan A, B, dan C.

b. Sifat distributif gabungan terhadap irisan

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
  ⇒ {a, b, c} ∪ {c, d}  = {a, b, c, d} ∩ {a, b, c, d, e}
  ⇒ {a, b, c, d} = {a, b, c, d}
  ⇒ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) berlaku.
• B ∪ (A ∩ C) = (B ∪ A) ∩ (B ∪ C)
  ⇒ {b, c, d} ∪ {c} = {a, b, c, d} ∩ {b, c, d, e}
  ⇒ {b, c, d} = {b, c, d}
  ⇒ B ∪ (A ∩ C) = (B ∪ A) ∩ (B ∪ C) berlaku.
• C ∪ (A ∩ B) = (C ∪ A) ∩ (C ∪ B)
  ⇒ {c, d, e} ∪ {b, c} = {a, b, c, d, e} ∩ {b, c, d, e}
  ⇒ {b, c, d, e} = {b, c, d, e}
  ⇒ C ∪ (A ∩ B) = (C ∪ A) ∩ (C ∪ B) berlaku.

KESIMPULAN

∴  Sifat distributif gabungan terhadap irisan berlaku pada himpunan A, B, dan C.