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Transcript of Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar...
![Page 1: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/1.jpg)
Teknik Digital 2011
![Page 2: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/2.jpg)
Pendahuluan Komputer digital modern dirancang, dipelihara, dan
operasinya dianalisis dengan memakai teknik dan simbologi dari bidang matematika yang dinamakan aljabar modern atau aljabar Boolean
pengetahuan mengenai aljabar boolean ini merupakan suatu keharusan dalam bidang komputer.
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KONSEP POKOK ALJABAR BOOLEAN
Variabel – variabel yang dipakai dalam persamaan aljabar boolean memiliki karakteristik
Variabel tersebut hanya dapat mengambil satu harga dari dua harga yang mungkin diambil. Kedua harga ini dapat dipresentasikan dengan simbol “ 0 ” dan “ 1 ”.
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Penambahan Logis 0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
![Page 5: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/5.jpg)
Perkalian Logis 0 . 0 = 0
0 . 1 = 0
1 . 0 = 0
1 . 1 = 1
![Page 6: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/6.jpg)
Komplementasi atau Negasi
0 = 1
1 = 0
![Page 7: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/7.jpg)
HUKUM DASAR ALJABAR BOOLEAN
a. Hukum Komutatif
A + B = B + A
A . B = B . A
b. Hukum Asosiatif
(A + B) + C = A + (B + C)
(A . B) . C = A . (B . C)
c. Hukum Distributif
A . (B + C) = A . B + A . C
A + (B . C) = (A + B) . ( A + C )
![Page 8: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/8.jpg)
d. Hukum Identitas
A + A = A
A . A = A
e. Hukum Negasi
(A) = A
A = A
f. Hukum Redundan
A + A . B = A
A . (A + B) = A
![Page 9: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/9.jpg)
g. Indentitas
0 + A = A
1 . A = A
1 + A = 1
0 . A = 0
A + A . B = A + B
i. Teorema De Morgan
(A + B) = A . B
(A . B) = A + B
![Page 10: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/10.jpg)
Resume Aljanar Boole(A1) X = 0 if X ¹ 1 (A1’) X = 1 if X ¹ 0
(A2) If X = 0, then X’ = 1 (A2’) if X = 1, then, X’ = 0
(A3) 0 . 0 = 0 (A3’) 1 + 1 = 1
(A4) 1 . 1 = 1 (A4’) 0 + 0 = 0
(A5) 0 . 1 = 1 . 0 = 0 (A5’) 1 + 0 = 0 + 1 = 1
(T1) X + 0 = X (T1’) X . 1 = X (Identities)
(T2) X + 1 = 1 (T2’) X . 0 = 0 (Null elements)
(T3) X + X = X (T3’) X . X = X (Idempotency)
(T4) (X’)’ = X (Involution)
(T5) X + X’ = 1 (T5’) X . X’ = 0 (Complements)
(T6) X + Y = Y + X (T6’) X . Y = Y . X (Commutativity)
(T7) (X + Y) + Z = X + (Y + Z) (T7’) (X . Y) . Z = X . (Y . Z) (Associativity)
(T8) X . Y + X . Z = X . (Y + Z) (T8’) (X + Y) . (X + Z) = X + Y . Z (Distributivity)
(T9) X + X . Y = X (T9’) X . (X + Y) = X (Covering)
(T10) X . Y + X . Y’ = X (T10’) (X + Y) . (X + Y’) = X (Combining)
(T11) X . Y + X’. Z + Y . Z = X . Y + X’ . Z
(T11’) (X + Y) . ( X’ + Z) . (Y + Z) = (X + Y) . (X’ + Z) (Consensus)
(T12) X + X + . . . + X = X (T12’) X . X . . . . . X = X (Generalized idempotency)
(T13) (X1 . X2 . . . . . Xn)’ = X1’ + X2’ + . . . + Xn’
(T13’) (X1 + X2 + . . . + Xn)’ = X1’ . X2’ . . . . . Xn’ (DeMorgan’s theorems)
(T14) [F(X1, X2, . . ., Xn, +, .)]’ = F(X1’, X2’, . . ., Xn’, . , +) (Generalized DeMorgran’s theorem)
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Summary 0 + X = X
1 + X = 1
X + X = X
X + X = 1
0 . X = 0
1 . X = X
X . X = X
X . X = 0
X = X
X + Y = Y + X
X . Y = Y . X
X + (Y + Z) = (X + Y) + Z
X . (Y . Z) = (X . Y) Z
X . (Y + Z) = XY + XZ
X + XZ = X
X (X + Y) = X
(X + Y) ( X + Z) = X + YZ
X + XY = X + Y
XY + YZ + YZ = XY + Z
![Page 12: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/12.jpg)
ContohSederhanakan ungkapan serta tabel kebenarannya
di bawah ini :
(X+Y) (X + Z)
Hasil := X + XZ + XY + YZ= X + XY + XZ + YZ= X (1+Y) + Z (X + Y)= X+Z (X+Y)= X + XZ + YZ= X (1+Z) + YZ= X + YZ
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PENGANTAR GERBANG LOGIKA Arsitektur sistem komputer tersusun atas rangkaian
logika 1 (true) dan 0 (false) yang dikombinasikan dengan sejumlah gerbang logika yaitu AND, OR, NOT, NOR, XOR, NAND.
Program komputer berjalan diatas dasar struktur penalaran yang baik dari suatu solusi terhadap suatu permasalahan dengan bantuan komponen program yaitu if-then, if – then –else dan lainnya.
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Gerbang NOT
![Page 15: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/15.jpg)
Gerbang AND
![Page 16: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/16.jpg)
Gerbang OR
![Page 17: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/17.jpg)
Gerbang NAND
![Page 18: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/18.jpg)
Gerbang NOR
![Page 19: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/19.jpg)
Gerbang XOR
![Page 20: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/20.jpg)
Gerbang XNOR
![Page 21: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/21.jpg)
Contoh
Carilah persamaan booleannya dan jika diketahuinilai inputan A dan B tinggi (1) dan yang nilaiinputan yang lain rendah (0) maka cari nilai hasilkeluarannya ?
![Page 22: Teknik Digital 2011 - cyber.unissula.ac.idcyber.unissula.ac.id/DIRBA/member/210603026/file/Aljabar boole.pdf · KONSEP POKOK ALJABAR BOOLEAN Variabel –variabel yang dipakai dalam](https://reader034.fdocuments.net/reader034/viewer/2022052300/5c85d3b309d3f2ea4b8d2690/html5/thumbnails/22.jpg)
Selesai