Logika biner (aljabar boolean, gerbang logika)
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Transcript of Logika biner (aljabar boolean, gerbang logika)
![Page 1: Logika biner (aljabar boolean, gerbang logika)](https://reader033.fdokumen.com/reader033/viewer/2022050703/5589499dd8b42a5f648b45f9/html5/thumbnails/1.jpg)
Logika BinerNugroho Adi Pramono
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terdiri
variabel biner
operasi logika
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Variabel Biner
A, B, C
x, y, z
punya dua (dan hanya dua) kemungkinan nilai
0, 1
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Operasi Logika
AND
OR
NOT
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AND
x . y = z
x AND y is equal to z
x DAN y sama dengan z
xy = z
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OR
x + y = z
x OR y is equal to z
x ATAU y sama dengan z
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NOT
x’ = z
NOT x is equal to z
BUKAN x sama dengan z
(operasi komplemen)
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Logika Biner ≠ Aritmatika Biner
1 + 1 = 10 -> satu tambah satu sama dengan dua(aritmatika)
1 + 1 = 1 -> satu ATAU satu sama dengan satu (logika)
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Tabel Kebenaran
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Gerbang Logika
Rangkaian elektronik
Beberapa input
Satu output
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Gerbang Logika
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Gerbang AND
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Gerbang AND
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Gerbang AND
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Gerbang AND
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Gerbang OR
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Gerbang OR
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Gerbang OR
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Gerbang OR
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Gerbang NOT
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Gerbang NOT
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The Timing Diagram
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Bagaimana timing-diagram-nya
jika inputnya lebih dari dua
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Aljabar Boolean
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Definisi
S adalah himpunan
x, y adalah obyek
x ∈ S artinya x anggota S
y ∉ S artinya y bukan elemen S
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Definisi
A = [1, 2, 3, 4]
Elemen himpunan A adalah angka 1, 2, 3, 4
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Operator Biner
a * b = c
* adalah operator biner
untuk mendapatkan c dari pasangan (a, b)
syarat a,b,c ∈ S
* bukan operator biner jika a,b ∈ S dan c ∉ S
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Postulat
Closure
Associative Law
Commutative Law
Identity Element
Inverse
Distributive Law
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Closure
Closure, tertutup
untuk setiap a, b ∈ N
selalu ada c ∈ N
yang memenuhi a + b = c
N tidak tertutup jika menggunakan operator -
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Associative Law
( x * y ) * z = x * ( y * z )
untuk semua x, y, z, ∈ S
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Commutative Law
x * y = y * x
untuk semua x, y ∈ S
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Identity Element
e * x = x * e = x untuk setiap x ∈ S
x + 0 = 0 + x = x untuk setiap x ∈ I
himpunan N tidak punya elemen identitas
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Identity Element
e * x = x * e = x untuk setiap x ∈ S
x + 0 = 0 + x = x untuk setiap x ∈ I
himpunan N tidak punya elemen identitas
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Inverse
jika S punya elemen identitas e
maka x ∈ S dikatakan punya invers y ∈ S
jika memenuhi x * y = e
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Distributive Law
x * ( y . z ) = ( x * y ) . ( x * z )
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Gunakan timing diagram !Bagaimanakah f dan g?
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Gunakan timing diagram !Bagaimanakah f dan g?
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Axioma
himpunan B
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Axioma
bersifat tertutup untuk operator + dan .
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Axioma
0 adalah elemen identitas untuk +
1 adalah elemen identitas untuk .
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Axioma
operator + bersifat komutatif
x + y = y + x
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Axioma
operator . bersifat komutatif
x . y = y . x
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Axioma
operator + bersifat distributif
x . ( y + z ) = ( x . y ) + ( x . z )
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Axioma
operator .bersifat distributif
x + ( y . z ) = ( x + y ) . ( x + z )
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Axioma
untuk setiap x ∈ B
terdapat x’ ∈ B (komplemen)
sehingga x + x’ = 1
dan x . x’ = 0
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Axioma
terdapat setidaknya dua elemen
x, y ∈ B
yang memenuhi x ≠ y
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“Hati-hati terhadap sifat distributif”
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Dualitas
kita dapat menukar OR dan AND dengan mengganti 0 dengan 1 atau sebaliknya
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Dualitas
x + 0 = x
x . 1 = x
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Dualitas
x + 1 = 1
x . 0 = 0
![Page 51: Logika biner (aljabar boolean, gerbang logika)](https://reader033.fdokumen.com/reader033/viewer/2022050703/5589499dd8b42a5f648b45f9/html5/thumbnails/51.jpg)
Dan dia hidup bahagia selama-lamanya...
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F = A + B
Dan dia hidup bahagia selama-lamanya...
A B F
0 0 0
0 1 1
1 0 1
1 1 1
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F = AB + B
Dan dia hidup bahagia selama-lamanya...
A B AB F
0 0 0 0
0 1 0 0
1 0 0 1
1 1 1 1
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F = A + BC
Dan dia hidup bahagia selama-lamanya...
A B C BC F
0 0 0 0 0
0 0 1 0 0
0 1 0 0 0
0 1 1 1 1
1 0 0 0 1
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
![Page 55: Logika biner (aljabar boolean, gerbang logika)](https://reader033.fdokumen.com/reader033/viewer/2022050703/5589499dd8b42a5f648b45f9/html5/thumbnails/55.jpg)
–Johnny Appleseed
“Type a quote here.”
Dan dia hidup bahagia selama-lamanya...
![Page 56: Logika biner (aljabar boolean, gerbang logika)](https://reader033.fdokumen.com/reader033/viewer/2022050703/5589499dd8b42a5f648b45f9/html5/thumbnails/56.jpg)
Selesai
Dan dia hidup bahagia selama-lamanya...