Teknik Penyederhanaan Karnaugh Map (K-Map)
TEKNIK PENYEDERHANAAN KARNAUGH
MAP (K-MAP)
3
Digital Logic
1
B
A
00
01
11
Digital Logic
10
11
B.C
10
B.C
1
A
A
Empat Variabel
CD
AB
1
B
0
A
Dua Variabel
Tiga Variabel
0
B
BC
00
B.C
01
B.C
0
A
1
A
00
C.D
01
C.D
11
C.D
10
C.D
SEDERHANAKAN FUNGSI 2 VARIABEL
F= A.B + A.B +
A.B
Hasil dari Kmap =
F= A + B
Digital Logic
A
B
B
B
0
1
A
0
1
1
1
0
1
A
PENYEDERHANAAN FUNGSI DARI TABEL KEBENARAN 2
VARIABEL
A
B
F
0
0
0
0
1
1
1
0
1
1
1
0
Hasil dari K-MAP :
Digital Logic
B
0
B’
1
B
0
A’
0
1
1
A
1
0
A
F = A’ B + A B’
SOAL :
1. F= A.B + A.B + A.B
2. F= A.B + A.B + A.B
3. F= A.B + A.B + A.B
Digital Logic
SEDERHANAKAN FUNGSI 3 VARIABEL
F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
A
00
BC
01
BC
11
BC
10
BC
0
A
0
1
1
1
1
A
1
1
0
0
BC
F= B.C + A.B +
A.B
Digital Logic
F= A.C + A.B +
A.B
ATAU
A
00
BC
01
BC
11
BC
10
BC
0
A
0
1
1
1
1
A
1
1
0
0
BC
SOAL :
1. F=
A.B.C
2. F=
A.B.C
3. F=
A.B.C
4. F=
A.B.C
5. F=
A.B.C
Digital Logic
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
PENYEDERHANAAN FUNGSI DARI TABEL KEBENARAN 3
VARIABEL
A
B
C
F
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
A
00
BC
01
BC
11
BC
10
BC
0
A
0
0
1
1
1
A
0
1
1
1
BC
Hasil dari K-MAP :
Digital Logic
F=B+AC
A
B
C
F1
F2
0
0
0
1
0
0
0
1
1
1
0
1
0
0
0
0
1
1
0
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
1
1
1
1
A
BC
01
BC
11
BC
10
BC
0
A
1
A
A
BC
0
A
1
A
Digital Logic
00
BC
00
BC
01
BC
11
BC
10
BC
SEDERHANAKAN FUNGSI 4 VARIABEL
F= A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D
+ A.B.C.D + A.B.C.D
00
C.D
01
C.D
11
C.D
10
C.D
00
A.B
0
1
1
0
01
A.B
1
1
1
1
11
A.B
0
1
1
0
10
A.B
0
1
0
0
CD
AB
Digital Logic
F= C.D + A.D + A.B + B.D
SOAL :
1. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
2. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
3. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
Digital Logic
PENYEDERHANAA
N FUNGSI DARI
TABEL KEBENARAN
4 VARIABEL
A
B
C
D
F
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
1
1
1
0
1
0
0
1
0
1
0
1
0
0
1
1
0
1
00
A.B
0
1
1
1
1
1
0
0
0
1
01
A.B
1
0
0
1
0
11
A.B
1
0
1
0
1
1
0
1
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
0
0
1
1
1
1
0
CD
AB
10
A.B
Digital Logic
00
C.D
01
C.D
11
C.D
10
C.D
A
B
C
D
F1
F2
0
0
0
0
1
1
0
0
0
1
1
1
0
0
1
0
1
0
01
A.B
0
0
1
1
1
1
0
1
0
0
0
0
11
A.B
0
1
0
1
1
1
0
1
1
0
0
0
0
1
1
1
0
0
1
0
0
0
1
0
1
0
0
1
1
0
1
0
1
0
1
1
1
0
1
1
1
0
01
A.B
1
1
0
0
0
1
1
1
0
1
0
1
11
A.B
1
1
1
0
0
0
1
1
1
1
0
1
CD
AB
00
C.D
01
C.D
11
C.D
10
C.D
00
A.B
10
A.B
CD
AB
00
A.B
10
Digital Logic
00
C.D
01
C.D
11
C.D
10
C.D
TUGAS : TULIS TANGAN DI KERTAS
FOLIO
1. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D
2. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D
3. F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
4. F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
Digital Logic
SOP (SUM OF PRODUCT)
Penjumlahan dari suatu fungsi
perkalian.
Contoh : F = (A.B.C) + (A.B.C) +
(A.B.C)
Dapat ditulis dengan F(A,B,C)=∑
{0,3,4}
Yang disederhanakan adalah angka ‘1’.
Digital Logic
CONTOH :
Sederhanakan
F(A,B,C,D)=∑
{0,1,3,4,7,8,10,11,15}
00
01
11
10
CD
AB
00
A.B
01
A.B
B
C
D
F
No
0
0
0
0
1
0
0
0
0
1
1
1
0
0
1
0
0
2
0
0
1
1
1
3
0
1
0
0
1
4
0
1
0
1
0
5
0
1
1
0
0
6
0
1
1
1
1
7
C.D
C.D
C.D
C.D
0
1
3
2
1
1
1
0
1
0
0
0
1
8
4
5
7
6
1
0
0
1
0
9
1
0
1
0
1
10
1
0
1
1
1
11
1
1
0
0
0
12
1
1
0
1
0
13
1
1
1
0
0
14
1
1
1
1
0
15
1
0
1
0
11
A.B
12
13
15
14
10
A.B
8
9
11
10
0
1
1
Digital Logic
A
0
1
0
1
0
POS (PRODUCT OF SUM)
Perkalian dari suatu fungsi
penjumlahan.
Contoh : F = (A+B+C) . (A+B+C) .
(A+B+C)
Dapat ditulis dengan F(A,B,C)=∏
{0,3,4}
Yang disederhanakan adalah angka ‘0’.
Digital Logic
CONTOH :
Sederhanakan F(A,B,C,D) =
∏{0,1,2,3,6,7,10,11,13,15}
A
B
C
D
F
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
2
0
0
1
1
0
3
0
1
0
0
1
4
0
1
0
1
1
5
0
1
1
0
0
6
0
1
1
1
0
7
1
0
0
0
1
8
1
0
0
1
1
9
1
0
1
0
0
10
00
C.D
01
C.D
11
C.D
10
C.D
0
1
3
2
0
0
0
0
4
5
7
6
1
1
0
0
11
A.B
12
13
15
14
1
1
0
1
1
0
11
10
A.B
8
9
11
10
1
1
0
0
1
12
1
1
0
1
0
13
1
1
1
0
1
14
1
1
1
1
0
15
CD
AB
00
A.B
01
A.B
1
1
0
1
0
0
F= (A’+B’).(A’+C).(A+B+C).
(A+B’+C)
Digital Logic
0
SOAL :
Sederhanakan
F(A,B,C,D) = ∏{0,1,3,4,5,10,12,14,15}
Sederhanakan
F(A,B,C,D) = ∏{1,2,5,6,8,11,12,14}
Sederhanakan
F(A,B,C,D) = ∏{0,2,3,5,6,8,9,11,13,15}
Digital Logic
DON’T CARE
Don’t care (d)
adalah kondisi
mengambang
dari suatu logika,
bisa dijadikan ke
logika ‘1’ atau ‘0’.
Contoh diberikan
tabel kebenaran
sbb:
Digital Logic
A
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
d
0
0
1
1
0
0
1
0
0
d
0
1
0
1
1
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
0
0
d
1
1
0
1
1
1
1
1
0
d
1
1
1
1
d
CD
AB
00
A.B
01
A.B
00
C.D
01
C.D
11
C.D
10
C.D
0
1
3
2
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
d
0
0
1
1
0
0
1
0
0
d
1
1
0
d
0
1
0
1
1
4
5
7
6
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
0
0
d
1
1
0
1
1
1
1
1
0
d
1
1
1
1
d
d
1
0
0
11
A.B
12
13
15
14
d
1
d
d
10
A.B
8
9
11
10
1
0
0
1
F = C’
Digital Logic
A
SOAL :
Sederhanakan F(A,B,C,D) =
∏{0,1,3,4,5,14,15} & d{ 2,11,13}
Sederhanakan F(A,B,C,D)=
∑ {0,1,3,4,7,8,11,15} &
d{ 4,10,12,13}
Digital Logic
MAP (K-MAP)
3
Digital Logic
1
B
A
00
01
11
Digital Logic
10
11
B.C
10
B.C
1
A
A
Empat Variabel
CD
AB
1
B
0
A
Dua Variabel
Tiga Variabel
0
B
BC
00
B.C
01
B.C
0
A
1
A
00
C.D
01
C.D
11
C.D
10
C.D
SEDERHANAKAN FUNGSI 2 VARIABEL
F= A.B + A.B +
A.B
Hasil dari Kmap =
F= A + B
Digital Logic
A
B
B
B
0
1
A
0
1
1
1
0
1
A
PENYEDERHANAAN FUNGSI DARI TABEL KEBENARAN 2
VARIABEL
A
B
F
0
0
0
0
1
1
1
0
1
1
1
0
Hasil dari K-MAP :
Digital Logic
B
0
B’
1
B
0
A’
0
1
1
A
1
0
A
F = A’ B + A B’
SOAL :
1. F= A.B + A.B + A.B
2. F= A.B + A.B + A.B
3. F= A.B + A.B + A.B
Digital Logic
SEDERHANAKAN FUNGSI 3 VARIABEL
F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
A
00
BC
01
BC
11
BC
10
BC
0
A
0
1
1
1
1
A
1
1
0
0
BC
F= B.C + A.B +
A.B
Digital Logic
F= A.C + A.B +
A.B
ATAU
A
00
BC
01
BC
11
BC
10
BC
0
A
0
1
1
1
1
A
1
1
0
0
BC
SOAL :
1. F=
A.B.C
2. F=
A.B.C
3. F=
A.B.C
4. F=
A.B.C
5. F=
A.B.C
Digital Logic
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C + A.B.C + A.B.C + A.B.C +
PENYEDERHANAAN FUNGSI DARI TABEL KEBENARAN 3
VARIABEL
A
B
C
F
0
0
0
0
0
0
1
0
0
1
0
1
0
1
1
1
1
0
0
0
1
0
1
1
1
1
0
1
1
1
1
1
A
00
BC
01
BC
11
BC
10
BC
0
A
0
0
1
1
1
A
0
1
1
1
BC
Hasil dari K-MAP :
Digital Logic
F=B+AC
A
B
C
F1
F2
0
0
0
1
0
0
0
1
1
1
0
1
0
0
0
0
1
1
0
0
1
0
0
0
1
1
0
1
0
0
1
1
0
1
1
1
1
1
1
1
A
BC
01
BC
11
BC
10
BC
0
A
1
A
A
BC
0
A
1
A
Digital Logic
00
BC
00
BC
01
BC
11
BC
10
BC
SEDERHANAKAN FUNGSI 4 VARIABEL
F= A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D
+ A.B.C.D + A.B.C.D
00
C.D
01
C.D
11
C.D
10
C.D
00
A.B
0
1
1
0
01
A.B
1
1
1
1
11
A.B
0
1
1
0
10
A.B
0
1
0
0
CD
AB
Digital Logic
F= C.D + A.D + A.B + B.D
SOAL :
1. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
2. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
3. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D
Digital Logic
PENYEDERHANAA
N FUNGSI DARI
TABEL KEBENARAN
4 VARIABEL
A
B
C
D
F
0
0
0
0
1
0
0
0
1
0
0
0
1
0
1
0
0
1
1
1
0
1
0
0
1
0
1
0
1
0
0
1
1
0
1
00
A.B
0
1
1
1
1
1
0
0
0
1
01
A.B
1
0
0
1
0
11
A.B
1
0
1
0
1
1
0
1
1
0
1
1
0
0
0
1
1
0
1
0
1
1
1
0
0
1
1
1
1
0
CD
AB
10
A.B
Digital Logic
00
C.D
01
C.D
11
C.D
10
C.D
A
B
C
D
F1
F2
0
0
0
0
1
1
0
0
0
1
1
1
0
0
1
0
1
0
01
A.B
0
0
1
1
1
1
0
1
0
0
0
0
11
A.B
0
1
0
1
1
1
0
1
1
0
0
0
0
1
1
1
0
0
1
0
0
0
1
0
1
0
0
1
1
0
1
0
1
0
1
1
1
0
1
1
1
0
01
A.B
1
1
0
0
0
1
1
1
0
1
0
1
11
A.B
1
1
1
0
0
0
1
1
1
1
0
1
CD
AB
00
C.D
01
C.D
11
C.D
10
C.D
00
A.B
10
A.B
CD
AB
00
A.B
10
Digital Logic
00
C.D
01
C.D
11
C.D
10
C.D
TUGAS : TULIS TANGAN DI KERTAS
FOLIO
1. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D
2. F=A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D + A.B.C.D + A.B.C.D + A.B.C.D +
A.B.C.D
3. F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
4. F= A.B.C + A.B.C + A.B.C + A.B.C +
A.B.C
Digital Logic
SOP (SUM OF PRODUCT)
Penjumlahan dari suatu fungsi
perkalian.
Contoh : F = (A.B.C) + (A.B.C) +
(A.B.C)
Dapat ditulis dengan F(A,B,C)=∑
{0,3,4}
Yang disederhanakan adalah angka ‘1’.
Digital Logic
CONTOH :
Sederhanakan
F(A,B,C,D)=∑
{0,1,3,4,7,8,10,11,15}
00
01
11
10
CD
AB
00
A.B
01
A.B
B
C
D
F
No
0
0
0
0
1
0
0
0
0
1
1
1
0
0
1
0
0
2
0
0
1
1
1
3
0
1
0
0
1
4
0
1
0
1
0
5
0
1
1
0
0
6
0
1
1
1
1
7
C.D
C.D
C.D
C.D
0
1
3
2
1
1
1
0
1
0
0
0
1
8
4
5
7
6
1
0
0
1
0
9
1
0
1
0
1
10
1
0
1
1
1
11
1
1
0
0
0
12
1
1
0
1
0
13
1
1
1
0
0
14
1
1
1
1
0
15
1
0
1
0
11
A.B
12
13
15
14
10
A.B
8
9
11
10
0
1
1
Digital Logic
A
0
1
0
1
0
POS (PRODUCT OF SUM)
Perkalian dari suatu fungsi
penjumlahan.
Contoh : F = (A+B+C) . (A+B+C) .
(A+B+C)
Dapat ditulis dengan F(A,B,C)=∏
{0,3,4}
Yang disederhanakan adalah angka ‘0’.
Digital Logic
CONTOH :
Sederhanakan F(A,B,C,D) =
∏{0,1,2,3,6,7,10,11,13,15}
A
B
C
D
F
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
2
0
0
1
1
0
3
0
1
0
0
1
4
0
1
0
1
1
5
0
1
1
0
0
6
0
1
1
1
0
7
1
0
0
0
1
8
1
0
0
1
1
9
1
0
1
0
0
10
00
C.D
01
C.D
11
C.D
10
C.D
0
1
3
2
0
0
0
0
4
5
7
6
1
1
0
0
11
A.B
12
13
15
14
1
1
0
1
1
0
11
10
A.B
8
9
11
10
1
1
0
0
1
12
1
1
0
1
0
13
1
1
1
0
1
14
1
1
1
1
0
15
CD
AB
00
A.B
01
A.B
1
1
0
1
0
0
F= (A’+B’).(A’+C).(A+B+C).
(A+B’+C)
Digital Logic
0
SOAL :
Sederhanakan
F(A,B,C,D) = ∏{0,1,3,4,5,10,12,14,15}
Sederhanakan
F(A,B,C,D) = ∏{1,2,5,6,8,11,12,14}
Sederhanakan
F(A,B,C,D) = ∏{0,2,3,5,6,8,9,11,13,15}
Digital Logic
DON’T CARE
Don’t care (d)
adalah kondisi
mengambang
dari suatu logika,
bisa dijadikan ke
logika ‘1’ atau ‘0’.
Contoh diberikan
tabel kebenaran
sbb:
Digital Logic
A
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
d
0
0
1
1
0
0
1
0
0
d
0
1
0
1
1
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
0
0
d
1
1
0
1
1
1
1
1
0
d
1
1
1
1
d
CD
AB
00
A.B
01
A.B
00
C.D
01
C.D
11
C.D
10
C.D
0
1
3
2
B
C
D
F
0
0
0
0
1
0
0
0
1
1
0
0
1
0
d
0
0
1
1
0
0
1
0
0
d
1
1
0
d
0
1
0
1
1
4
5
7
6
0
1
1
0
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
1
1
0
1
0
0
1
0
1
1
0
1
1
0
0
d
1
1
0
1
1
1
1
1
0
d
1
1
1
1
d
d
1
0
0
11
A.B
12
13
15
14
d
1
d
d
10
A.B
8
9
11
10
1
0
0
1
F = C’
Digital Logic
A
SOAL :
Sederhanakan F(A,B,C,D) =
∏{0,1,3,4,5,14,15} & d{ 2,11,13}
Sederhanakan F(A,B,C,D)=
∑ {0,1,3,4,7,8,11,15} &
d{ 4,10,12,13}
Digital Logic