DIGITAL LOGIC

DIGITAL LOGIC

Engineer S. A. AHSAN RAJONLecturer

Department of Computer Science,Khulna Public College, Khulna.

E-mail: [email protected]

mailto:[email protected]

Engineer S. A. AHSAN RAJON KHULNA PUBLIC COLLEGE, KHULNA

TRUE AND FALSE !!! Rahim will get A+ Karim will get A+

Rahim will not get A+ Karim will not get A+

Rahim Will Get A+ Karim will not A+

Rahim will not get A+ Karim will get A+


TRUE AND FALSE => 1 & 0 !!! Rahim will get A+ >> TRUE Karim will get A+ >> TRUE

Rahim will get A+ >> FALSE Karim will get A+ >> TRUE

Rahim Will Get A+ >> TRUE Karim will get A+ >> FALSE

Rahim will get A+ >> FALSE Karim will get A+ >> FALSE


TRUE AND FALSE => 1 & 0 !!! Rahim will get A+ TRUE >> 1

Karim will get A+ TRUE >> 1

Rahim will get A+ FALSE >> 0 Karim will get A+ TRUE >> 1

Rahim Will Get A+ TRUE >> 1

Karim will get A+ FALSE >> 0

Rahim will get A+ FALSE >> 0

Karim will get A+ FALSE >> 0


TRUE AND FALSE => 1 & 0 !!!

RAHIM will get A+

RAHIM will get

A+

1 1

1 0

0 1

0 0


TRUE AND FALSE => 1 & 0 !!! Thus we may define everything with

simply 0 AND 1

What computer understands is simply TRUE and FALSE.

So, computations based on this binary is sufficient for any computations.


LOGIC GATES We may also have all the computations

based on a couple of operations. AND OR NOT

NAND NOR


AND AND

TRUE AND TRUE >> TRUE TRUE AND FALSE >> FALSE FALSE AND TRUE >> FALSE FALSE AND FALSE >> FALSE

It meansA B ANDed

RESULT

1 1 1

1 0 0

0 1 0

0 0 0


OR OR

TRUE AND TRUE >> TRUE TRUE AND FALSE >> TRUE FALSE AND TRUE >> TRUE FALSE AND FALSE >> FALSE

It meansA B ORed RESULT

1 1 1

1 0 1

0 1 1

0 0 0


NOT NOT

NOT of TRUE >> FALSE NOT of FALSE >> TRUE

It means

A NOT

1 0

0 1


NOT GATE

A A


REPRESENTATION AND represented by . Often A^B is also used.

OR Represented by A+B

NOT Represented by A’


FOR THREE VARIABLES Rahim will get A+ Karim will get A+ Mohim will get A+

Rahim will not get A+ Karim will not get A+ Mohim will get A+

Rahim Will Get A+ Karim will not A+ Mohim will get A+

Rahim will not get A+ Karim will not get A+ Mohim will get A+


FOR THREE VARIABLES Rahim will get A+ Karim will get A+ Mohim will not get A+

Rahim will not get A+ Karim will not get A+ Mohim will not get A+

Rahim Will Get A+ Karim will not A+ Mohim will not get A+

Rahim will not get A+ Karim will not get A+ Mohim will not get A+


FOR THREE VARIABLES Rahim will get A+ >> TRUE >> 1 Karim will get A+ >> TRUE >> 1 Mohim will get A+ >> TRUE >> 1

Rahim will get A+ >> FALSE >> 0 Karim will get A+ >> TRUE >> 1 Mohim will get A+ >> TRUE >> 1

Rahim Will Get A+ >> TRUE >> 1 Karim will get A+ >> FALSE >> 0 Mohim will get A+ >> TRUE >> 1

Rahim will get A+ >> FALSE >> 0 Karim will get A+ >> FALSE >> 0 Mohim will get A+ >> TRUE >> 1


FOR THREE VARIABLES Rahim will get A+ >> TRUE >> 1 Karim will get A+ >> TRUE >> 1 Mohim will get A+ >> FALSE >> 0

Rahim will get A+ >> FALSE >> 0 Karim will get A+ >> TRUE >> 1 Mohim will get A+ >> FALSE >> 0

Rahim Will Get A+ >> TRUE >> 1 Karim will get A+ >> FALSE >> 0 Mohim will get A+ >> FALSE >> 0

Rahim will get A+ >> FALSE >> 0 Karim will get A+ >> FALSE >> 0 Mohim will get A+ >> FALSE >> 0


FOR THREE VARIABLES

RAHIM will get A+

RAHIM will get A+

Mohim willl get A+

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1


OR OF THREE VARIABLES

X Y ZX OR Y OR

Zi.e. X+Y+Z

0 0 0 0

0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1


OR OR

TRUE AND TRUE >> TRUE TRUE AND FALSE >> TRUE FALSE AND TRUE >> TRUE FALSE AND FALSE >> FALSE

It meansA B OR-ed RESULT

1 1 1

1 0 1

0 1 1

0 0 0


AND OF THREE VARIABLES

X Y ZX AND Y AND Z

i.e. X.Y.Z

0 0 0 0

0 0 1 0

0 1 0 0

0 1 1 0

1 0 0 0

1 0 1 0

1 1 0 0

1 1 1 1


NOT NOT

NOT of TRUE >> FALSE NOT of FALSE >> TRUE

It means

A NOT

1 0

0 1


LOGIC GATES… THESE ARE CALLED GATES

AND GATE OR GATE NOT GATE


TWO INPUT AND GATE

O/PA

B


THREE INPUT AND GATE

O/PA

CB


TWO INPUT OR GATE

O/PA

B


TWO INPUT OR GATE

O/PA

cB


THREE INPUT AND GATE

O/PA

CB


OTHER GATES NAND GATES

NAND = NOT OF AND

NOR GATE NOT OF OR


THREE INPUT NAND GATE

O/PA

CB


THREE INPUT NOR GATE

O/PA

cB


NAND OF THREE VARIABLES

X Y ZAND OF X Y

Zi.e. (X.Y.Z)

NOT(AND OF X Y Z)_____

(X.Y.Z)

NAND (X,Y,Z)

0 0 0 0 1 1

0 0 1 0 1 1

0 1 0 0 1 1

0 1 1 0 1 1

1 0 0 0 1 1

1 0 1 0 1 1

1 1 0 0 1 1

1 1 1 1 0 0


NOR OF THREE VARIABLES

X Y Z X OR Y OR Z

NOT OF

(X+Y+Z)

NOR (A,B,C)

0 0 0 0 1

0 0 1 1 0

0 1 0 1 0

0 1 1 1 0

1 0 0 1 0

1 0 1 1 0

1 1 0 1 0

1 1 1 1 0


XOR Exclusive OR Gate

When both the operands are same, the result is 0 When the operands are different, the result is 1.

A B XORed RESULT

0 0 0

0 1 1

1 0 1

1 1 0

BABA B A


XOR Exclusive OR Gate


XNOR GATE… XNOR = NOT of XOR A XNOR B = NOT OF (A XOR B) =

BABA B XNORA

BABA B A

B A


XNOR GATE… XNOR

A B XOR XNOR

0 0 0 1

0 1 1 0

1 0 1 0

1 1 0 1


CODES ASCII

American Standard Codes for Information Interchange

128 Then 256 English and Latin


CODES BCD

Binary Coded Decimal 4 bits 8421 Code BCD 1001=(8X1)+(4X0)+(2X0)+(1X1) = DECIMAL

9 BCD 0010=(8X0)+(4X0)+(2X1)+(1X0) = DECIMAL

2


CODES… EBCDIC

Extended Binary Coded Decimal Interchange Format.

For example: 00000101=Pound Sign


CODES UNICODE

Sign/Symbol for all the languages. Now commonly used. Unicode Consortium. 65,535+ codes


AGAIN BOOLEAN ALGEBRA


REVISITING GATES…


LOGIC GATES….


SIMPLE MATH…

C__C

__A+B+ C

___________ __

A+B+ CAB


De Morgan’s theorem:

and,

De Morgan’s theorem:

and,


DE MORGAN’S THEOREM…

X Y Z __X

__Y

_Z

X+Y+Z

_____X+Y+

Z

X.Y.Z

____X.Y.Z

_ _ _X+Y+Z

_ _ _X .Y.

Z

0 0 0 1 1 1 0 1 0 1 1 1

0 0 1 1 1 0 1 0 0 1 1 0

0 1 0 1 0 1 1 0 0 1 1 0

0 1 1 1 0 0 1 0 0 1 1 0

1 0 0 0 1 1 1 0 0 1 1 0

1 0 1 0 1 0 1 0 0 1 1 0

1 1 0 0 0 1 1 0 0 1 1 0

1 1 1 0 0 0 1 0 1 0 0 0


BINARY ARITHMETIC…


BOOLEAN ALGEBRA…

Other important things:


BOOLEAN ALGEBRA… Simplification


BOOLEAN ALGEBRA… Slight Change:


BOOLEAN ALGEBRA…

We have already proved the above… The equivalent logic circuit has also been

designed. Since, those two are same, it is the same to draw

the simplified circuit.


BOOLEAN ALGEBRA…

__A

A

LOGIC CIRCUIT OF

__ __ A. B. C

__B

B

C

__ __ A. B. C


DECIMAL TO BINARY…… (123.445)10 = (?)2


BINARY TO DECIMAL…… (1001011.1100111)2=(?)10


BINARY CALCULATION…… BINARY SUMMATION

1+1=0 (SUM) 1+1=1 (CARRY)

0+0=0 0+1=1 1+0=1


BINARY CALCULATION…… BINARY subtraction

1-1=0 (SUM) 1+1=1 (CARRY)

0-0=0 0-1=1


BINARY SUMMATION 1100 1000 ______________________ 10100


BINARY SUMMATION 10100.01 11000.11 ___________________ 101101.00


BINARY SUBTRACTION 11001 1001 _______________ 10000


BINARY SUBTRACTION 101001.00 11001.11 __________________ 1111.01

0-1=1; Have a borrow from the left of next digit. This borrow will be added with the left of

next.


BINARY COMPLEMENT COMPLEMENT OF 0 is 1 COMPLEMENT OF 1 is 0


SUBTRACTION USING COMPLEMENT

9-4=5

WITHOUT COMPLEMENT

9 = 1001 -4 = -0100 __________________ 101


SUBTRACTION USING COMPLEMENT

9-4=5

WITH COMPLEMENTS

9 = 1001-4 = 1011 [4= 0100; So, 1’s complement of 4 is 1011 ]

__________________ 1 0100Since, there is carry, there should have 1 plus with the main result .

Thus, The result is 0100 +1 ____________ 0101

Since 5=0101, it is proved.

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DIGITAL LOGIC