Sistem Digital : Aljabar Boolean, Hukum Aljabar Boolean, dan Manipulasi Boolean, dan Contoh Soal
Contemporary Multilevel Machines
Level terendah ialah level digital logic, objek yang menarik di level ini ialah gates. Setiap gate mempunyai satu atau lebih digital inputs (masukan digital) yang me- representasikan 0 atau 1).
Masukan tersebut akan dihitung dan menjadi output dengan beberapa operasi/fungsi ibarat OR, AND, dst.
Aljabar Boolean
Operasi komputer didasarkan pada penyimpanan dan pemrosesan data biner. Elemen penyimpanan (exist pada satu dari dua state) dan circuit sanggup mengoperasikan data biner membentuk banyak sekali fungsi computer
Implementasi elemen storage dan circuit dalam digital logic, khususnya combinational dan sequential circuit.
Boolean Algebra is a Mathematical discipline used to design and analyze the behavior of the digital circuitry in digital computers and other digital systems. Named after George Boole which he is an English mathematician. He proposed basic principles of the algebra in 1854.
Later, Claude Shannon suggested Boolean algebra could be used to solve problems in relay- switching circuit design
Boolean Algebra is a Mathematical discipline used to design and analyze the behavior of the digital circuitry in digital computers and other digital systems. Named after George Boole which he is an English mathematician. He proposed basic principles of the algebra in 1854.
Later, Claude Shannon suggested Boolean algebra could be used to solve problems in relay- switching circuit design
Is a convenient tool :
Analysis
It is an economical way of describing the function of digital circuitry
Design
Given a desired function, Boolean algebra can be applied to develop a simplified implementation of that function
Boolean Variables and Operations
Menggunakan variable dan operasi
- Logic
- A variable may take on the value 1 (TRUE) or 0 (FALSE)
- Basic logical operations are AND, OR, and NOT
AND
- Yields true (binary value 1) if and only if both of its operands are true
- In the absence of parentheses the AND operation takes precedence over the OR operation
- When no ambiguity will occur the AND operation is represented by simple concatenation instead of the dot operator
OR
- Yields true if either or both of its operands are true
NOT
- Inverts the value of its operand
Simbol Operator
Simbol untuk mepresentasikan operasi
- AND = .
- OR = +
- NOT = overbar
Contoh
- A AND B = A . B
- A OR B = A + B
- NOT A = 𝐴
Evaluasi lisan Boolean
- 0 + 0 + 0 + 0 =
- 0 + 0 + 0 + 1 =
- 1 + 1 + 1 + 1 =
- 1.1 + 0.0 + 1 =
- 1.0.0.1 =
- 1.0 + 1.0 + 0.1 + 0.1 =
Contoh
Jika B = 0 dan C = 1; A = 1
- 𝐷 = 𝐴 + (𝐵 . 𝐶) ?
- 𝐷 = (𝐴 + 𝐵 ) . 𝐶 ?
- 𝐷 = 𝐴 + 𝐵 . 𝐵 + 𝐶 ?
- 𝐷 = 𝐵 + 𝐶 + 𝐴. 𝐵 ?
Operasi Boolean dari dua variable
Truth Tables / Tabel Kebenaran
XOR (exclusive –OR) operand logic = 1 bila dan hanya bila salah satu operand = 1
NAND : complement (NOT) dari fungsi AND
NOR : complement (NOT) dari fungsi OR
A NAND B : NOT (A AND B) = AB
A NOR B : NOT (A OR B) = A + B
Basic Identities of Boolean Algebra
Contoh
Terapkan teorema De Morgan pada lisan berikut :
( 𝑋𝑌𝑍 )' = ?
𝑋' + 𝑌' + 𝑍' = ?
((A+BC’)’ + D(E+F’)’)’ =
((A+BC’)’ + D(E+F’)’)’ =
Truth Tables / Tabel Kebenaran
Verifikasi Teorema DeMorgan
Teorema DeMorgan untuk 3 atau lebih variabel
Truth Table
A + AB = A
Buat Truth Table dari dari fungsi Boolean berikut :
X = AB + C’D
Contoh Persamaan/Fungsi Aljabar Boolean
- X + X’ .Y = (X + X’).(X +Y) = X+Y
- X .(X’+Y) = X.X’ + X.Y = X.Y
- X.Y+ X’.Z+Y.Z = X.Y + X’.Z + Y.Z.(X+X’)
- = X.Y + X’.Z + X.Y.Z + X’.Y.Z
- = X.Y.(1+Z) + X’.Z.(1+Y)
- = X.Y + X’.Z
Manipulasi Aljabar Boolean
Aljabar Boolean merupakan tool mempunyai kegunaan untuk menyederhanakan rangkaian digital (circuit digital). Perhatikan persamaan Boolean berikut :
Disederhanakan memakai operasi dasar aljabar boolean
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