*Sekarang kita telah mengetahui konsep bilangan biner, dan kita akan mempelajari cara menggambarkan bagaimana sistem menggunakan menggunakan level logika biner dalam membuat keputusan.
* Aljabar Boolean adalah alat yang penting dalam menggambarkan, menganalisa, merancang, dan mengimplementasikan rangkaian digital.
Konstanta Boolean dan Variabel.
Aljabar Boolean dibawah ini hanya mempunyai dua nilai : 0 dan 1.
Logika 0 dapat dikatakan : false, off, low, no, saklar terbuka.
Logika 1 dapat dikatakan: true, on, high, yes, saklar tertutup.
Tiga operasi logika dasar: OR, AND, dan NOT.
Tabel Kebenaran
Sebuah tabel kebenaran menggambarkan hubungan antara input dan ouput sebuah rangkaian logika.
Jumlah The number of entries corresponds to the number of inputs. For example a 2 input table would have 2 2 = 4 entries. A 3 input table would have 2 3 = 8 entries.
Contoh tabel kebenaran dengan masukan 2, 3 dan 4 buah.
Ÿ The Boolean expression for the OR operation is
X = A + B
Ÿ This is read as “x equals A or B.”
Ÿ X = 1 when A = 1 or B = 1.
Ÿ Truth table and circuit symbol for a two input OR gate:
OR Operation With OR Gates
Ÿ The OR operation is similar to addition but when A = 1 and B = 1, the OR operation produces 1 + 1 = 1.
Ÿ In the Boolean expression
x=1+1+1=1
We could say in English that x is true (1) when A is true
(1) OR B is true (1) OR C is true (1).
Ÿ There are many examples of applications where an output function is desired when one of multiple inputs is activated.
AND Operations with AND gates
Ÿ The Boolean expression for the AND operation is
X = A • B
Ÿ This is read as “x equals A and B.”
Ÿ x = 1 when A = 1 and B = 1.
Truth table and circuit symbol for a two input AND gate are shown. Notice the difference between OR and AND gates
Ÿ The AND operation is similar to multiplication.
Ÿ In the Boolean expression
X = A • B • C
X = 1 only when A = 1, B = 1, and C = 1.
NOT Operation
Ÿ The Boolean expression for the NOT operation is
X = A
Ÿ This is read as:
Ÿ x equals NOT A, or
Ÿ x equals the inverse of A, or
Ÿ x equals the complement of A
Ÿ Truth table, symbol, and sample waveform for the NOT circuit.
Describing Logic Circuits Algebraically
Ÿ The three basic Boolean operations (OR, AND, NOT) can describe any logic circuit.
Ÿ If an expression contains both AND and OR gates the AND operation will be performed first, unless there is a parenthesis in the expression.
Ÿ Examples of Boolean expressions for logic circuits:
Ÿ The output of an inverter is equivalent to the input with a bar over it. Input A through an inverter equals A.
Ÿ Examples using inverters.
Evaluating Logic Circuit Outputs
Ÿ Rules for evaluating a Boolean expression:
Ÿ Perform all inversions of single terms.
Ÿ Perform all operations within parenthesis.
Ÿ Perform AND operation before an OR operation unless parenthesis indicate otherwise.
Ÿ If an expression has a bar over it, perform the operations inside the expression and then invert the result.
Ÿ Evaluate Boolean expressions by substituting values and performing the indicated operations:
A = 0, B = 1, C = 1, and D = 1 x = ABC(A + D)
= 0 ×1×1× (0 +1)
= 1×1×1× (0 +1)
= 1×1×1× (1)
= 1×1×1× 0
= 0
Ÿ Output logic levels can be determined directly from a circuit diagram.
Ÿ The output of each gate is noted until a final output is found.
Implementing Circuits From Boolean Expressions
Ÿ It is important to be able to draw a logic circuit from a Boolean expression.
Ÿ The expression
x = A ×B×C
could be drawn as a three input AND gate.
Ÿ A more complex example such as
y = AC + BC + ABC
could be drawn as two 2-input AND gates and one 3-input AND gate feeding into a 3-input OR gate. Two of the AND gates have inverted inputs.
NOR Gates and NAND Gates
Ÿ Combine basic AND, OR, and NOT operations.
Ÿ The NOR gate is an inverted OR gate. An inversion “bubble” is placed at the output of the OR gate.
The Boolean expression is,
Ÿ The NAND gate is an inverted AND gate. An inversion “bubble” is placed at the output of the AND gate.
Ÿ The Boolean expression is
x = AB
Ÿ The output of NAND and NOR gates may be found by simply determining the output of an AND or OR gate and inverting it.
Ÿ The truth tables for NOR and NAND gates show the complement of truth tables for OR and AND gates.
Universality of NAND and NOR Gates
Ÿ NAND or NOR gates can be used to create the three basic logic expressions (OR, AND, and INVERT)
Ÿ This characteristic provides flexibility and is very useful in logic circuit design.
IEEE/ANSI Standard Logic Symbols
Ÿ Compare the IEEE/ANSI symbols to traditional symbols.
Ÿ These symbols are not widely accepted but may appear in some schematics.
Application
Summary of Methods to Describe Logic Circuits
Ÿ The three basic logic functions are AND, OR, and NOT.
Ÿ Logic functions allow us to represent a decision process.
Ÿ If it is raining OR it looks like rain I will take an umbrella.
Ÿ If I get paid AND I go to the bank I will have money to spend.
sumber : https://onlinelearning.uhamka.ac.id
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