03Gerbang Logika & aljabar boolean - REGULATIONpakendy.weebly.com/uploads/2/4/5/6/2456272/... · Konstanta Boolean dan Variabel. Aljabar Boolean dibawah ini hanya mempunyai dua nilai

Gerbang Logika dan

Aljabar Boolean

Program Studi T. Elektro

FT - UHAMKA

Slide - 3 1

ENDY SAENDY SAProgram Studi Teknik Elektro

Fakultas Teknik

Universitas Muhammadiyah Prof. Dr. HAMKA

Gerbang Logika dan Aljabar

Boolean

� Sekarang kita telah mengetahui konsep

bilangan biner, dan kita akan mempelajari

cara menggambarkan bagaimana sistem

menggunakan menggunakan level logika


FT - UHAMKA

Slide - 3 2

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.


FT - UHAMKA

Slide - 3 3

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.


FT - UHAMKA

Slide - 3 4

� Jumlah The number of entries corresponds to

the number of inputs. For example a 2 input

table would have 22 = 4 entries. A 3 input

table would have 23 = 8 entries.

Tabel Kebenaran

� Contoh tabel kebenaran dengan masukan 2, 3 dan

4 buah.


FT - UHAMKA

Slide - 3 5

Operasi OR dengan gerbang OR

� The Boolean expression for the OR operation is

X = A + B

� This is read as “x equals A or B.”

� X = 1 when A = 1 or B = 1.


FT - UHAMKA

Slide - 3 6

� Truth table and circuit symbol for a two input OR

gate:

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


FT - UHAMKA

Slide - 3 7

� In the Boolean expressionx=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.


FT - UHAMKA

Slide - 3 8

AND Operations with AND gates

� The Boolean expression for the AND operation is

X = A • B

� This is read as “x equals A and B.”

� x = 1 when A = 1 and B = 1.

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


FT - UHAMKA

Slide - 3 9

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

shown. Notice the difference between OR and AND gates.

Operation With AND Gates

� The AND operation is similar to multiplication.

� In the Boolean expressionX = A • B • C

X = 1 only when A = 1, B = 1, and C = 1.


FT - UHAMKA

Slide - 3 10

X = 1 only when A = 1, B = 1, and C = 1.

NOT Operation

� The Boolean expression for the NOT operation is

AX =


FT - UHAMKA

Slide - 3 11

� This is read as:

� x equals NOT A, or

� x equals the inverse of A, or

� x equals the complement of A

AX =

NOT Operation

� Truth table, symbol, and sample waveform

for the NOT circuit.


FT - UHAMKA

Slide - 3 12

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


FT - UHAMKA

Slide - 3 13

gates the AND operation will be performed

first, unless there is a parenthesis in the

expression.


Algebraically

� Examples of Boolean expressions for logic

circuits:


FT - UHAMKA

Slide - 3 14


Algebraically

� 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.


FT - UHAMKA

Slide - 3 15

� Examples using inverters.

Evaluating Logic Circuit Outputs

� Rules for evaluating a Boolean expression:

� Perform all inversions of single terms.

� Perform all operations within parenthesis.

� Perform AND operation before an OR operation


FT - UHAMKA

Slide - 3 16

� Perform AND operation before an OR operation

unless parenthesis indicate otherwise.

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

operations inside the expression and then invert

the result.


� Evaluate Boolean expressions by substituting

values and performing the indicated

operations:

1D and 1,C 1,B 0,A ====


FT - UHAMKA

Slide - 3 17

0

0111

)1(111

1)(0111

1)(0110

D)(ABCAx

1D and 1,C 1,B 0,A

=

⋅⋅⋅=

⋅⋅⋅=

+⋅⋅⋅=

+⋅⋅⋅=

+=

====


� Output logic levels can be determined directly

from a circuit diagram.

The output of each gate is noted until a final


FT - UHAMKA

Slide - 3 18

� 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

CBAx ⋅⋅=


FT - UHAMKA

Slide - 3 19

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.

BCACBACy ++=

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


FT - UHAMKA

Slide - 3 20

inversion “bubble” is placed at the output

of the OR gate.

� The Boolean expression is,BAx +=


� The NAND gate is an inverted AND gate.

An inversion “bubble” is placed at the

output of the AND gate.

The Boolean expression is


FT - UHAMKA

Slide - 3 21

� The Boolean expression is

ABx =


� 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


FT - UHAMKA

Slide - 3 22

� 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


FT - UHAMKA

Slide - 3 23

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


FT - UHAMKA

Slide - 3 24

Combinations of NANDs are used to create the three logic functions.


FT - UHAMKA

Slide - 3 25

combinations of NORs are used to create the three logic functions.

The Exclusive OR BAX ⊕=


FT - UHAMKA

Slide - 3 26

The Exclusive NOR BAABX +=


FT - UHAMKA

Slide - 3 27

IEEE/ANSI Standard Logic

Symbols

� Compare the

IEEE/ANSI symbols

to traditional symbols.

� These symbols are


FT - UHAMKA

Slide - 3 28

� These symbols are

not widely accepted

but may appear in

some schematics.

Application


FT - UHAMKA

Slide - 3 29

Application


FT - UHAMKA

Slide - 3 30

Exercise


FT - UHAMKA

Slide - 3 31

Output

Boolean expressions ???

Truth Table???

Exercise

V2

0V

V1

0V L1

U6A

U5A

U4A

U1A


FT - UHAMKA

Slide - 3 32

V4

0V

V3

0V

U6B

U4B L2

U3A

U2A

Boolean expressions ???

Truth Table???

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.


FT - UHAMKA

Slide - 3 33

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.

Thank You


FT - UHAMKA

Slide - 3 34

Documents

03Gerbang Logika & aljabar boolean - REGULATIONpakendy.weebly.com/uploads/2/4/5/6/2456272/... · Konstanta Boolean dan Variabel. Aljabar Boolean dibawah ini hanya mempunyai dua nilai