"TUGAS 3: RANGKUMAN GERBANG LOGIKA"

Gerbang Logika dan Aljabar Boolean

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:

o    A2 input table would have 2^2 = 4 entries.

o    A3 input table would have 2^3 = 8 entries..

Contoh tabel kebenaran dengan masukan 2, 3 dan 4 buah.

o  Masukan 2 buah

A	B		X
0 0 1 1	0 1 0 1		1 0 1 0

o  Masukan 3 buah

A	B	C		X
0 0 0 0 1 1 1 1	0 0 1 1 0 0 1 1	0 1 0 1 0 1 0 1		0 1 1 0 0 0 0 1

o Masukan 4 buah

A	B	C	D		X
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1	0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1	0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1	0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1		0 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1

Operasi OR dengan gerbang OR

The Boolean expression for the OR operation is

o  X = A + B

o  This is read as “x equals A or B.”

o  X = 1 when A = 1 or B = 1.

Truth table and circuit symbol for a two input OR gate:

           OR                                                                

A	B		X = A + B
0 0 1 1	0 1 0 1		0 1 1 1

OR Operation With OR Gates

The OR operation is similar to addition but when A = 1 and B = 1, the OR operation produces 1 + 1 = 1.

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).

OR Operation With OR Gates

There are many examples of applications where an output function is desired when one of multiple inputs is activated.

AND Operations with AND gates

The Boolean expression for the AND operation is

X = A • B

o  This is read as “x equals A and B.”

o  x = 1 when A = 1 and B = 1.

Truth table and circuit symbol for a two input AND gate are

Truth table and circuit symbol for a two input AND gate are

shown. Notice the difference between OR and AND gates.

                    AND

A	B		X = A . B
0 0 1 1	0 1 0 1		0 0 0 1

 

NOT Operation

The Boolean expression for the NOT operation is

X = A

This is read as:

o  x equals NOT A, or

o  x equals the inverse of A, or

o  x equals the complement of A

Truth table, symbol, and sample waveform for the NOT circuit.

Describing Logic Circuits Algebraically

The three basic Boolean operations (OR, AND, NOT) can describe any logic circuit.

If an expression contains both AND and OR gates the AND operation will be performed first, unless there is a parenthesis in the expression.

Examples of Boolean expressions for logic circuits:

The output of an inverter is equivalent to the input with a bar over it. Input A through an inverter equals A.

Examples using inverters:

Evaluating Logic Circuit Outputs

Rules for evaluating a Boolean expression:

o  Perform all inversions of single terms.

o  Perform all operations within parenthesis.

o  Perform AND operation before an OR operation

o  Perform AND operation before an OR operation unless parenthesis indicate otherwise.

o  If an expression has a bar over it, perform the

o  operations inside the expression and then invert the result.

Evaluate Boolean expressions by substituting values and performing the indicated operations:

Output logic levels can be determined directly from a circuit diagram.

The output of each gate is noted until a final output is found.

Implementing Circuits From Boolean Expressions

It is important to be able to draw a logic circuit from a Boolean expression.

The expression:

X = A⋅B⋅C

could be drawn as a three input AND gate.

A more complex example such as:

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

Combine basic AND, OR, and NOT operations.

The NOR gate is an inverted OR gate. An inversion “bubble” is placed at the output of the OR gate.

The Boolean expression is,

The NAND gate is an inverted AND gate. An inversion “bubble” is placed at the output of the AND gate.

The Boolean expression is,

The output of NAND and NOR gates may be found by simply determining the output of an AND or OR gate and inverting it.

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

NAND or NOR gates can be used to create the three basic logic expressions (OR, AND, and INVERT)

This characteristic provides flexibility and is very useful in logic circuit design.

IEEE/ANSI Standard Logic Symbols

Compare the IEEE/ANSI symbols to traditional symbols.

These symbols are not widely accepted but may appear in some schematics.

Application

Summary of Methods to Describe Logic Circuits

The three basic logic functions are AND, OR, and NOT.

Logic functions allow us to represent a decision process.

If it is raining OR it looks like rain I will take an umbrella.

If I get paid AND I go to the bank I will have money to spend.

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