1. dedi26-belajar.blogspot.com 2012
LATIHAN SOAL ALJABAR BOOLE
Latihan soal Bab 5
1. Guanakan sifat-sifat aljabar Boole untuk membuktikan identitas 𝑎 ∗ 𝑎′ ⊕ 𝑏 = 𝑎 ∗ 𝑏
Penyelesaian:
Pembuktian ruas kiri
𝑎 ∗ 𝑎′ ⊕ 𝑏
 𝑎 ∗ 𝑎′ ⊕ (𝑎 ∗ 𝑏) (D1)
 0 ⊕ (𝑎 ∗ 𝑏) (C1)
 (𝑎 ∗ 𝑏) (3B)
Terbukti
2. Dalam suatu aljabar Boole, perhatikan bahwa 𝑎 = 𝑏 ⟺ 𝑎 ∗ 𝑏′ ⊕ 𝑎′ ∗ 𝑏 = 0
Penyelesaian:
 𝑎 ∗ 𝑏′ ⊕ 𝑎′ ∗ 𝑏 = 0
 𝑎 ∗ 𝑏′ ⊕ 𝑎′ ∗ [(𝑎 ∗ 𝑏′ ) ⊕ 𝑏] = 0 (D2)
′ ′ ′ ′
[ 𝑎⊕ 𝑎 ∗ 𝑏 ⊕ 𝑎 ) ∗ 𝑎⊕ 𝑏 ∗ 𝑏 ⊕ 𝑏 =0 (D2)
′ ′
 [1 ∗ 𝑏 ⊕ 𝑎 ) ∗ 𝑎 ⊕ 𝑏 ∗ 1 = 0 (C1)
Karena 𝑎 = 𝑏, maka
 𝑏′ ⊕ 𝑏′ ∗ (𝑏 ⊕ 𝑏) = 0 (B3)
 𝑏′ ∗ 𝑏 = 0 (1L)
0=0 (C1)
Terbukti
3. Sederhanakan ekspersi Boole berikut 𝑎 ∗ 𝑏 ′ ⊕ 𝑎 ⊕ 𝑏 ′
Penyelesaian:
 𝑎∗ 𝑏 ′⊕ 𝑎⊕ 𝑏 ′
 (𝑎′ ⊕ 𝑏′ ) ⊕ (𝑎′ ∗ 𝑏′ ) (C3) dan (3C)
 (𝑎′ ⊕ 𝑏′ ) ⊕ 𝑎′ ∗ [ 𝑎′ ⊕ 𝑏′ ⊕ 𝑏′ ] (D2)
 𝑎′ ⊕ 𝑎′ ⊕ 𝑎′ ⊕ 𝑏′ ∗ [ 𝑎′ ⊕ 𝑏′ ⊕ 𝑏′ ⊕ 𝑏′ ] (D2)
 𝑎′ ⊕ 𝑎′ ⊕ 𝑏′ ∗ [ 𝑎′ ⊕ 𝑏′ ⊕ 𝑏′ ] (1L)
 𝑎′ ⊕ 𝑎′ ⊕ 𝑏′ ∗ [𝑎′ ⊕ 𝑏′ ⊕ 𝑏′ ] (3L)
 𝑎′ ⊕ 𝑏′ ∗ (𝑎′ ⊕ 𝑏′ ) (1L)
 (𝑎′ ⊕ 𝑏′ ) (L1)
 (𝑎 ∗ 𝑏)′ (C3)
Jadi bentuk sederhana dari 𝑎 ∗ 𝑏 ′ ⊕ 𝑎 ⊕ 𝑏 ′ = (𝑎 ∗ 𝑏)′
Aljabar Boole dedi26 Page 1

2. dedi26-belajar.blogspot.com 2012
4. Tunjukan bahwa dalam sebuah aljabar Boole berlaku 𝑎 ∗ 𝑏 ⊕ 𝑎 ∗ 𝑏′ = 𝑎
Penyelesaian:
 𝑎 ∗ 𝑏 ⊕ 𝑎 ∗ 𝑏′ = 𝑎
 𝑎 ∗ 𝑏 ⊕ 𝑎 ∗ 𝑎 ∗ 𝑏 ⊕ 𝑏′ = 𝑎 (D2)
 𝑎 ⊕ 𝑎 ∗ 𝑏 ⊕ 𝑎 ∗ 𝑎 ⊕ 𝑏′ ∗ 𝑏 ⊕ 𝑏′ = 𝑎 (D2)
 𝑎 ∗ 𝑏 ⊕ 𝑎 ∗ 𝑎 ⊕ 𝑏′ ∗ 1 = 𝑎 (1L) dan (1C)
 𝑎 ∗ 𝑎 ⊕ 𝑏′ = 𝑎 (L4) dan (B3)
 𝑎 ∗ 𝑎 ⊕ 𝑎 ∗ 𝑏′ = 𝑎 (D1)
 𝑎⊕0= 𝑎 (L1) dan (P3)
 𝑎= 𝑎 (3B)
Terbukti
Latihan Soal Bab 6
1) Buktikan kesamaan berikut
′
𝑎 ∗ 𝑏′ ⊕ 𝑐 ∗ 𝑏′ ⊕ 𝑎 ∗ 𝑐 ′ ′
= 𝑎′
Penyelesaian:
′
 𝑎 ∗ 𝑏′ ⊕ 𝑐 ∗ 𝑏′ ⊕ 𝑎 ∗ 𝑐 ′ ′
= 𝑎′
 𝑎′ ⊕ 𝑏′ ⊕ 𝑐 ′
∗ 𝑏′ ⊕ 𝑎′ ⊕ 𝑐 = 𝑎′ (C3)
′ ′ ′ ′ ′
 𝑎 ⊕ 𝑏 ⊕ 𝑐 ∗ 𝑎 ⊕ 𝑏 ⊕ 𝑐 = 𝑎′ (3L)
Misalkan, 𝑑 = 𝑏′ ⊕ 𝑐. Maka:
 𝑎′ ⊕ 𝑑 ′ ∗ 𝑎′ ⊕ 𝑑 = 𝑎′
 𝑎′ ⊕ 𝑑 ′ ∗ 𝑎′ ⊕ 𝑎′ ⊕ 𝑑 ′ ∗ 𝑑 = 𝑎′ (D1)
 𝑎′ ∗ 𝑎′ ⊕ 𝑎′ ∗ 𝑑 ′ ⊕ [ 𝑎′ ∗ 𝑑 ⊕ 𝑑 ′ ∗ 𝑑 ] = 𝑎′ (D1)
 𝑎′ ⊕ 𝑎′ ∗ 𝑑 ′ ⊕ 𝑎′ ∗ 𝑑 ⊕ 0 = 𝑎′ (L1) dan (C1)
 𝑎′ ⊕ 𝑎′ ∗ 𝑑 = 𝑎′ (4L) dan (3B)
 𝑎′ = 𝑎′ (4L)
Terbukti
2) Sederhanakan ekspresi berikut
𝑎 ⊕ 𝑏 ∗ 𝑎′ ∗ 𝑏′
Penyelesaian:
 𝑎 ⊕ 𝑏 ∗ 𝑎′ ∗ 𝑏′
 𝑎⊕ 𝑏 ∗ 𝑎⊕ 𝑏 ′ (3C)
0 (C1)
Jadi bentuk sederhana dari 𝑎 ⊕ 𝑏 ∗ 𝑎′ ∗ 𝑏′ = 0
Aljabar Boole dedi26 Page 2

3. dedi26-belajar.blogspot.com 2012
3) Buatlah tabel evaluasi dari ekspresi berikut
𝑥 ⊕ 𝑦 ∗ 𝑥 ′ ⊕ 𝑧 ∗ (𝑦 ⊕ 𝑧)
Penyelesaian:
𝑥 𝑦 𝑧 (𝑥 ⊕ 𝑦) (𝑥 ′ ⊕ 𝑧) (𝑦 ⊕ 𝑧) (𝐴)
0 0 0 0 1 0 0
0 0 1 0 1 1 0
0 1 0 1 1 1 1
0 1 1 1 1 1 1
1 0 0 1 0 0 0
1 0 1 1 1 1 1
1 1 0 1 0 1 0
1 1 1 1 1 1 1
Aljabar Boole dedi26 Page 3