dedi26-belajar.blogspot.com 2012

Aljabar Boole dedi26 Page 1

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)′

dedi26-belajar.blogspot.com 2012

Aljabar Boole dedi26 Page 2

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

dedi26-belajar.blogspot.com 2012

Aljabar Boole dedi26 Page 3

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