Logic Gates & Boolean Algebra

Course: B.Sc.(CS)

Sem.:1st

Subject: Basics of Digital Electronics

Unit: 2nd

Basic Definitions

Binary Operators◦ AND

z = x • y = x y z=1 if x=1 AND y=1

◦ ORz = x + y z=1 if x=1 OR y=1

◦ NOTz = x = x’ z=1 if x=0

Boolean Algebra◦ Binary Variables: only ‘0’ and ‘1’ values

◦ Algebraic Manipulation

Boolean Algebra Postulates

Commutative Law

x • y = y • x x + y = y + x

Identity Element

x • 1 = x x + 0 = x

Complement

x • x’ = 0 x + x’ = 1

Boolean Algebra Theorems

Duality

◦ The dual of a Boolean algebraic expression is

obtained by interchanging the AND and the OR

operators and replacing the 1’s by 0’s and the 0’s

by 1’s.

◦ x • ( y + z ) = ( x • y ) + ( x • z )

◦ x + ( y • z ) = ( x + y ) • ( x + z )

Theorem 1

◦ x • x = x x + x = x

Theorem 2

◦ x • 0 = 0 x + 1 = 1

Applied to a

valid equation

produces a

valid equation

Boolean Algebra Theorems

Theorem 3: Involution◦ ( x’ )’ = x ( x ) = x

Theorem 4: Associative & Distributive◦ ( x • y ) • z = x • ( y • z )( x + y ) + z = x + ( y + z

)

◦ x • ( y + z ) = ( x • y ) + ( x • z )

x + ( y • z ) = ( x + y ) • ( x + z)

Theorem 5: De Morgan◦ ( x • y )’ = x’ + y’ ( x + y )’ = x’ • y’

◦ ( x • y ) = x + y ( x + y ) = x • y

Theorem 6: Absorption◦ x • ( x + y ) = x x + ( x • y ) = x

Boolean Functions

Boolean Expression

Example: F =

x + y’ z

Truth Table

All possible

combinations

of input variables

Logic Circuit

x y z F

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 1

1 1 0 1

1 1 1 1x

yz

F

Algebraic Manipulation

Literal:A single variable within a term that may be complemented or not.

Use Boolean Algebra to simplify Boolean functions to produce simpler circuitsExample: Simplify to a minimum number of literals

F = x + x’ y ( 3 Literals)

= x + ( x’ y )

= ( x + x’ ) ( x + y )

= ( 1 ) ( x + y ) = x + y ( 2 Literals)

Distributive law (+ over

•)

Complement of a Function

DeMorgan’s Theorm

Duality & Literal Complement

CBAF

CBAF

CBAF

CBAF

CBAF

CBAF

Canonical Forms

Minterm

◦ Product (AND function)

◦ Contains all variables

◦ Evaluates to ‘1’ for a

specific combination

Example

A = 0 A B C

B = 0 (0) • (0) • (0)

C = 01 • 1 • 1 = 1

A B C Minterm

0 0 0 0 m0

1 0 0 1 m1

2 0 1 0 m2

3 0 1 1 m3

4 1 0 0 m4

5 1 0 1 m5

6 1 1 0 m6

7 1 1 1 m7

Canonical Forms

Maxterm

◦ Sum (OR function)

◦ Contains all variables

◦ Evaluates to ‘0’ for a

specific combination

Example

A = 1 A B C

B = 1 (1) + (1) + (1)

C = 10 + 0 + 0 = 0

A B C Maxterm

0 0 0 0 M0

1 0 0 1 M1

2 0 1 0 M2

3 0 1 1 M3

4 1 0 0 M4

5 1 0 1 M5

6 1 1 0 M6

7 1 1 1 M7

Canonical Forms

Truth Table to Boolean Function

CBAF CBA CBA ABCA B C F

0 0 0 0

0 0 1 1

0 1 0 0

0 1 1 0

1 0 0 1

1 0 1 1

1 1 0 0

1 1 1 1

Canonical Forms

Sum of Minterms

Product of Maxterms

ABCCBACBACBAF

7541 mmmmF

)7,5,4,1(F

CABBCACBACBAF

CABBCACBACBAF

CABBCACBACBAF

))()()(( CBACBACBACBAF

6320 MMMMF

(0,2,3,6)F

Standard Forms Sum of Products (SOP)

ABCCBACBACBAF BA

BA

CCBA

)1(

)(

AC

BBAC

)(

CB

AACB

)(

)()()( BBACCCBAAACBF

ACBACBF

Standard Forms Product of Sums (POS)

CABBCACBACBAF

)( CCBA

)( AACB

)( BBCA

)()()( AACBCCBABBCAF

CBBACAF

))()(( CBBACAF

Two-Level Implementations Sum of Products (SOP)

Product of Sums (POS)

B’C

FB’A

AC

AC

FB’A

B’C

ACBACBF

))()(( CBBACAF

Logic Operators

AND

NAND (Not AND)

xy

x • y

xy

x • y

x y AND

0 0 0

0 1 0

1 0 0

1 1 1

x y NAND

0 0 1

0 1 1

1 0 1

1 1 0

Logic Operators

OR

NOR (Not OR)

xy

x + y

xy

x + y

x y OR

0 0 0

0 1 1

1 0 1

1 1 1

x y NOR

0 0 1

0 1 0

1 0 0

1 1 0

Logic Operators

XOR (Exclusive-OR)

XNOR (Exclusive-NOR)

(Equivalence)

xy

x Å yx y + x y

xy

x Å y

x � yx y + x y

x y XOR

0 0 0

0 1 1

1 0 1

1 1 0

x y XNOR

0 0 1

0 1 0

1 0 0

1 1 1

Logic Operators

NOT (Inverter)

Buffer

x x

x x

x NOT

0 1

1 0

x Buffer

0 0

1 1

Multiple Input Gates

De Morgan’s Theorem on

Gates AND Gate

◦ F = x • y F = (x • y) F = x +

y

OR Gate

◦ F = x + y F = (x + y) F = x • y

Change the “Shape” and “bubble” all lines

Universal Gates

NOR Gate as an Inverter

Gate

X XZ

XXX (Before Bubble)

X Z

0 1

1 0Equivalent to Inverter

NOR Gate as an OR Gate

X

YYX Y XZ

Y X

NOR Gate “Inverter”

X Y Z

0 0 0

0 1 1

1 0 1

1 1 1

Equivalent to OR Gate

NOR Gate as an AND Gate

Y XY X Y XZ

“Inverters” NOR Gate

X Y Z

0 0 0

0 1 0

1 0 0

1 1 1

Equivalent to AND Gate

NOR Gate Equivalent of AOI

Gates

AND OR INVERTER

1. If starting from a logic expression, implement the design with AOI logic.

2. In the AOI implementation, identify and replace every AND,OR, and INVERTER gate with its NOR equivalent.

3. Redraw the circuit.

4. Identify and eliminate any double inversions. (i.e. back-to-back inverters)

5. Redraw the final circuit.

Process for NOR Implementation

NAND Gate

X

YY X YXZ

X Y Z

0 0 1

0 1 1

1 0 1

1 1 0

NAND Gate as an Inverter

GateXXX

XZ

X Z

0 1

1 0 Equivalent to Inverter

NAND Gate as an AND Gate

X

Y YX YXZ

YX

NAND Gate Inverter

X Y Z

0 0 0

0 1 0

1 0 0

1 1 1

Equivalent to AND

Gate

NAND Gate as an OR Gate

X

Y

YXY X Y XZ

X

NAND GateInverters

Y

X Y Z

0 0 0

0 1 1

1 0 1

1 1 1

Equivalent to OR Gate

NAND Gate Equivalent to AOI

Gates

AND OR INVERTER

1. If starting from a logic expression, implement the design with AOI logic.

2. In the AOI implementation, identify and replace every AND,OR, and INVERTER gate with its NAND equivalent.

3. Redraw the circuit.

4. Identify and eliminate any double inversions (i.e., back-to-back inverters).

5. Redraw the final circuit.

Process for NAND

Implementation

References

Book References:

Digital Logic and Computer Design –

M. Morris Mano – Pearson

Fundamentals of Digital Circuits – A.

Anand Kumar – PHI

Modern Digital Electronics - R.P. Jain -

TMH

Digital Electronics -Tokneinh – MGH

Web References www.ddegjust.ac.in/studymaterial/pg

dcal

www.indianshout.com/digital-

electronics-notes-material/3023

www.wiley.com/college/engin/balabani

an293512/pdf/ch04.pdf

www.electronics-tutorials.ws

Image References:

http://www.youshouldgoaway.com/wp-

content/uploads/2014/11/thankyou.jpg