Jika A = {1,3,4,7,8,9,12} B = {1,2,3,5,7,8} C={2,4,6,8,10}

Jawaban bervariasi tergantung interpretasi operasi '+'. Mengasumsikan '+' adalah gabungan (U): i. {1, 3, 4, 5, 7, 8, 9, 12} ii. {1, 3, 4, 7, 8, 9, 12} iii. {1, 2, 3, 4, 5, 6, 7, 8, 10} iv. {2, 4, 6, 8, 9, 10, 11, 12, 13, 14} vi. {9, 12} vii. {11, 13, 14}

Pembahasan

Diberikan himpunan:
A = {1, 3, 4, 7, 8, 9, 12}
B = {1, 2, 3, 5, 7, 8}
C = {2, 4, 6, 8, 10}
S = {x | x adalah bilangan asli ≤ 14} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

a. Diagram Venn akan menunjukkan ketiga lingkaran (A, B, C) yang saling beririsan di dalam sebuah persegi panjang (S). Irisan antar himpunan dan elemen yang hanya ada di satu himpunan harus digambarkan dengan benar.

b. Perhitungan operasi himpunan:

i. A + (B - C)
   B - C = {1, 2, 3, 5, 7, 8} - {2, 4, 6, 8, 10} = {1, 3, 5, 7}
   A + (B - C) = {1, 3, 4, 7, 8, 9, 12} + {1, 3, 5, 7} = {1, 3, 4, 5, 7, 8, 9, 12}
   (Operasi '+' di sini diasumsikan sebagai gabungan atau penjumlahan elemen yang tidak jelas, namun jika diartikan sebagai gabungan:
A U (B - C) = {1, 3, 4, 7, 8, 9, 12} U {1, 3, 5, 7} = {1, 3, 4, 5, 7, 8, 9, 12})

ii. A ∩ (A U B)
    A U B = {1, 3, 4, 7, 8, 9, 12} U {1, 2, 3, 5, 7, 8} = {1, 2, 3, 4, 5, 7, 8, 9, 12}
    A ∩ (A U B) = {1, 3, 4, 7, 8, 9, 12} ∩ {1, 2, 3, 4, 5, 7, 8, 9, 12} = {1, 3, 4, 7, 8, 9, 12} = A
    (Sifat Idempoten: A ∩ (A U B) = A)

iii. B U (B + C)
     B + C = {2, 4, 6, 8, 10} + {2, 4, 6, 8, 10} = {2, 4, 6, 8, 10} (jika '+' berarti himpunan itu sendiri atau gabungan)
     B U (B + C) = {1, 2, 3, 5, 7, 8} U {2, 4, 6, 8, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 10}

iv. (B - C)^c + (A - B)
    (B - C)^c = S - (B - C) = {1, ..., 14} - {1, 3, 5, 7} = {2, 4, 6, 8, 9, 10, 11, 12, 13, 14}
    A - B = {1, 3, 4, 7, 8, 9, 12} - {1, 2, 3, 5, 7, 8} = {4, 9, 12}
    (B - C)^c U (A - B) = {2, 4, 6, 8, 9, 10, 11, 12, 13, 14} U {4, 9, 12} = {2, 4, 6, 8, 9, 10, 11, 12, 13, 14}

vi. (A U B) - (C + B)
    (A U B) = {1, 2, 3, 4, 5, 7, 8, 9, 12}
    C + B = {2, 4, 6, 8, 10} U {1, 2, 3, 5, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8, 10}
    (A U B) - (C U B) = {1, 2, 3, 4, 5, 7, 8, 9, 12} - {1, 2, 3, 4, 5, 6, 7, 8, 10} = {9, 12}

vii. (B + C)^c - A
     (B U C) = {1, 2, 3, 4, 5, 6, 7, 8, 10}
     (B U C)^c = S - (B U C) = {1, ..., 14} - {1, 2, 3, 4, 5, 6, 7, 8, 10} = {9, 11, 13, 14}
     (B U C)^c - A = {9, 11, 13, 14} - {1, 3, 4, 7, 8, 9, 12} = {11, 13, 14}