Soal-Soal Matematika/Himpunan

Elemen keanggotaan

Elemen keanggotaan dilambangkan sebagai ∈.

Himpunan semesta

Himpunan semesta dilambangkan sebagai S.

Himpunan kosong

Himpunan kosong dilambangkan sebagai {} atau ∅.

Himpunan bagian

Himpunan bagian dilambangkan sebagai ⊂. contoh:

1. A = {11, 12, 13, 14, 15}, B = {11, 13} dan C = {16, 17} maka B ⊂ A serta C ⊄ A.

Himpunan irisan

Himpunan irisan dilambangkan sebagai ∩. contoh:

1. A = {11, 12, 13, 14, 15}, B = {11, 12, 15, 16, 17} maka A ∩ B = {11, 12, 15}.

Himpunan gabungan

Himpunan irisan dilambangkan sebagai ∪. contoh:

1. A = {11, 12, 13, 14, 15}, B = {11, 12, 15, 16, 17} maka A ∪ B = {11, 12, 13, 14, 15, 16, 17}.

sifat

1. A ∩ B = B ∩ A
2. A ∪ B = B ∪ A
3. (A ∩ B) ∩ C = A ∩ (B ∩ C)
4. (A ∪ B) ∪ C = A ∪ (B ∪ C)
5. A ∩ (B ∪ C) = (A ∩ C) ∪ (B ∩ C)
6. A ∪ (B ∩ C) = (A ∪ C) ∩ (B ∪ C)
7. (A’)’ = A
8. A ∩ A’ = ∅
9. A ∪ A’ = S
10. A ∩ S = A
11. A ∪ S = S
12. A ∩ ∅ = ∅
13. A ∪ ∅ = A
14. (A ∩ B)’ = A’ ∪ B’
15. (A ∪ B)’ = A’ ∩ B’

Contoh:

• S = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, A = {x|11 ≤ x ≤ 15, x ∈ N} dan B = {x|14 ≤ x ≤ 18, x ∈ N}. Tentukan:
  • A’
  • B’
  • A ∩ B
  • A ∪ B
  • A’ ∩ B’
  • A’ ∪ B’
  • A’ ∩ B
  • A’ ∪ B

Jawaban

• S = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, A = {11, 12, 13, 14, 15} dan B = {14, 15, 16, 17, 18}

• • A’ = {16, 17, 18, 19, 20}
  • B’ = {11, 12, 13, 19, 20}
  • A ∩ B = {14, 15}
  • A ∪ B = {11, 12, 13, 14, 15, 16, 17, 18}
  • A’ ∩ B’ = {19, 20}
  • A’ ∪ B’ = {11, 12, 13, 16, 17, 18, 19, 20}
  • A’ ∩ B = {16, 17, 18}
  • A’ ∪ B = {14, 15, 16, 17, 18, 19, 20}

Keterangan

• (A ∩ B)’ = A’ ∪ B’
• (A ∪ B)’ = A’ ∩ B’
• (A ∩ B’)’ = A’ ∪ B
• (A ∪ B’)’ = A’ ∩ B