Diketahui S Bilangan Cacah Kurang Dari 15

a. P’ = {7, 8, 9, 10, 11, 12, 13, 14}

b. Q’ = {1, 2, 3, 4, 10, 11, 12, 13, 14}

c. P ∩ Q = {5, 6}

    P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9}

    (P ∩ Q)’ = {1, 2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14}

d. (P ∪ Q)’ = {10, 11, 12, 13, 14}

2). Diketahui
S = {bilangan cacah kurang dari 11}
A = {x | x ∈ P, x < 10, P bilangan prima}
B = {5, 7, 9}
Tentukan
a. Ac
b. Bc
c. (A ∩ B)c
d. (A ∪ B)c
e. A ∩ (A ∪ B)c
f. Bc ∩ (A ∪ B)
g. (A ∪ B)c ∩ (A ∪ B)c
h. (Ac ∩ B)c ∪ (A ∪ B c)c

Jawab :



S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {2, 3, 5, 7}

B = {5, 7, 9}

a. A’ = {0, 1, 4, 6, 8, 9, 10}.

b. B’ = {0, 1, 2, 3, 4, 6, 8, 10}.

c. (A ∩ B)’ = {0, 1, 2, 3, 4, 6, 8, 9, 10}.

    A ∪ B = {2, 3, 5, 7} ∪ {5, 7, 9} = {2, 3, 5, 7, 9}.

d. (A ∪ B)’ = {0, 1, 4, 6, 8, 10}.

e. A ∩ (A U B)’ = {2, 3, 5, 7} ∩ {0, 1, 4, 6, 8, 10} = ∅.

f. B’ ∩ (A ∪ B) = {0, 1, 2, 3, 4, 6, 8, 10} ∩ {2, 3, 5, 7, 9} = {2, 3}.

g. (A ∪ B)’ ∩ (A ∪ B)’ = {0, 1, 4, 6, 8, 10} ∩ {0, 1, 4, 6, 8, 10} = {0, 1, 4, 6, 8, 10}.

     A’ ∩ B = {0, 1, 4, 6, 8, 9, 10} ∩ {5, 7, 9} = {9}.

     (A’ ∩ B)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.

h. (A’ ∩ B)’ ∪ (A ∪ B’)’ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 10} ∪ {0, 1, 4, 6, 8, 10} = {0, 1, 4, 6, 8, 10}.

3). Perhatikan diagram Venn berikut ini

Berdasarkan diagram Venn tersebut tentukan banyak anggota dari
a. Ac
b. Bc
c. Cc
d. (A ∩ B)c
e. (A ∪ C)c
f. (A ∩ C)c
g. Ac ∩ (B ∪ C)c
h. (A ∩ B)c ∩ (A ∩ C)c

Jawab :



a. banyak anggota complemen A = 25 + 3 + 18 = 46

= 20 + 18 + 3 + 25 = 66

4). Diketahui A = {a, b, c, d, e, f} dan B = {e, f, g, h, j}. Tentukan
a. A – B
b. B – A
c. (A – B) ∩ A
d. (A – B) ∪ (B – A)

5). Misalkan A = {1, 2, 3, …, 10} dan B = {3, 5, 7, 9,11, 13}, dan
C = {7, 8, 9, 10, 11, 12, 13}. Tentukan anggota himpunan dari
a. A – B
b. B – A
c. B – C
d. C – A
e. (A – B) ∩ (A – C)
f. (A – C) ∪ (B – C)
g. (A ∪ B) – (B ∪ C)
h. (A – B)c ∪ (B – C)

Jawab :



a. A – B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {3, 5, 7, 9, 11, 13} = {1, 2, 4, 6, 8, 10}.

b. B – A = {3, 5, 7, 9, 11, 13} – {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = {11, 13}.

c. B – C = {3, 5, 7, 9, 11, 13} – {7, 8, 9, 10, 11, 12, 13} = {3, 5}

   C – B = {7, 8, 9, 10, 11, 12, 13} – {3, 5, 7, 9, 11, 13} = {8, 10, 12}

d. C – A = {7, 8, 9, 10, 11, 12, 13} – {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} = {11, 12, 13}

    A – C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} – {7, 8, 9, 10, 11, 12, 13} = {1, 2, 3, 4, 5, 6}

e. (A – B) ∩ (A – C) = {1, 2, 4, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6} = {1, 2, 4, 6}.

f. (A – C) ∪ (B – C) = {1, 2, 3, 4, 5, 6} ∪ {3, 5} = {1, 2, 3, 4, 5, 6}

    A U B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∪ {3, 5, 7, 9, 11, 13} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}

    B ∪ C = {3, 5, 7, 9, 11, 13} ∪ {7, 8, 9, 10, 11, 12, 13} = {3, 5, 7, 8, 9, 10, 11, 12, 13}

g. (A∪B)-(B∪C) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13}-{3, 5, 7, 8, 9, 10, 11, 12, 13} = {1, 2, 4, 6, 12}

Untuk menentukan komplemen, seharusnya diketahui himpunan semesta S. Kemungkinan S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} sehingga

h. (A – B)’ ∪ (B – C)’ = {3, 5, 7, 9, 11, 13} ∪ {1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} = S.