1. Digital Logic Chapter-2
Boolean Algebra and Logic Gates

2. Outline of Chapter -2 2.1 Introduction 2.2 Basic definitions
2.3 Axiomatic Definition of Boolean Algebra
2.4 Basic Theorems & Properties of Boolean
Algebra
2.5 Boolean Functions
2.6 Canonical and Standard Forms
-Minterms and Maxterms
2.7 Other Logic Operations
2.8 Digital Logic Gates
2.9 Integrated Circuits

3. Today’s Lecture Objectives
In this Lecture you will learn about Boolean algebra with
Theorems and Boolean Functions.
At the end of this Lecture you will be able to answer the
below questions which is in Text book exercise.
Define Boolean Algebra?
What is Boolean Operators?
Simplify the Boolean Functions in the exercise.

4. Boolean Algebra – Introduction Basic definitions
To define a Boolean algebra
The set B
Rules for two binary operations
The elements of B and rules should conform to our axioms
Two-valued Boolean algebra
B = {0, 1} x y
x · y 1 x y
x + y 1 x x’ 1

5. Basic Theorems & Properties of Boolean Algebra
x+0 = x
(a) x+x| = 1
x+x = x
(a) x+1 = 1
(x|)| = x
(a) x+y=y+x
x+(y+z)=(x+y)+z
(a) x(y+z)=xy+xz
(x+y)| = x| + y|
(a) x+xy=x
(b) x.1=x
(b) x. x| = 0
(b) x.x = x
(b) x.0 = 0
(b) xy = yx
(b)x(yz) = (xy)z
(b) x+yz = (x+y)(x+z)
(b) (xy)| = x| + y|
(b) x(x+y) = x

6. Operator Precedence Parentheses NOT AND OR Example: (x + y)’ x’ · y’
x + x · y’

7. Boolean Functions Consists of
binary variables (normal or complement form)
the constants, 0 and 1
logic operation symbols, “+” and “·”
Example:
F1(x, y, z) = x + y’ z
F2(x, y, z) = x’ y’ z + x’ y z + xy’ F2 F1 z y x 1 1 1 1 1 1 1 1

8. Logic Circuit Diagram of F1
F1(x, y, z) = x + y’ z x
x + y’ z y z
y’z
Gate Implementation of F1 = x + y’ z

9. Logic Circuit Diagram of F2
F2 = x’ y’ z + x’ y z + xy’ x y z F2
Algebraic manipulation
F2 = x’ y’ z + x’ y z + xy’
F2 = x’ z + xy’

10. Alternative Implementation of F2x y z F2
F2 = x’ y’ z + x’ y z + xy’ F2 x y z

11. Today’s Lecture Objectives
In this Lecture you will learn about Basic Logic Gates,
Universal Gates with Two Level implementation.
In this Lecture you will learn about Standard forms as Sum of
Min Terms and Max Terms with Integrated Circuits.
At the end of this Lecture you will be able to answer the
below questions which is in Text book exercise.
Design the Basic Logic Gates
Design the Universal Logic Gates
Simplify the Minterms for Boolean functions in the exercise.

12. Logic Gate Symbols
TRANSFER
NOT
AND OR
XOR
XNOR
NAND
NOR

13. Digital Logic Gates

14. Universal Gates NAND and NOR gates are universal
We know any Boolean function can be written in terms of three logic operations:
AND, OR, NOT
In return, NAND gate can implement these three logic gates by itself
So can NOR gate
(x’ y’ )’
y ’ x’
(xy)’ y x 1 1 1

15. NAND Gate x
NOT OR x y x y
AND

16. NOR Gate x x y x y

17. Canonical & Standard Forms
Minterms
A product term: all variables appear (either in its normal, x, or its complement form, x’)
How many different terms we can get with x and y?
x’y’ → 00 → m0
x’y → 01 → m1
xy’ → 10 → m2
xy → 11 → m3
m0, m1, m2, m3 (minterms or AND terms, standard product)
n variables can be combined to form 2n minterms

18. Canonical & Standard Forms
Maxterms (OR terms, standard sums)
M0 = x + y → 00
M1 = x + y’ → 01
M2 = x’ + y →10
M3 = x’ + y’ → 11
n variables can be combined to form 2n maxterms
m0’ = M0
m1’ = M1
m2’ = M2
m3’ = M3

19. Min- & Maxterms with n = 3 x y z Minterms Maxterms term designation
x’y’z’ m0
x + y + z M0 1
x’y’z m1
x + y + z’ M1
x’yz’ m2
x + y’ + z M2
x’yz m3
x + y’ + z’ M3
xy’z’ m4
x’ + y + z M4
xy’z m5
x’ + y + z’ M5
xyz’ m6
x’ + y’ + z M6
xyz m7
x’ + y’ + z’ M7 *

20. Important Properties
Any Boolean function can be expressed as a sum of minterms
Any Boolean function can be expressed as a product of maxterms
Example:
F’ = Σ (0, 2, 3, 5, 6) = x’y’z’ + x’yz’ + x’yz + xy’z + xyz’
How do we find the complement of F’?
F = (x + y + z)(x + y’ + z)(x + y’ + z’)(x’ + y + z’)(x’ + y’ + z) *

21. Canonical Form
If a Boolean function is expressed as a sum of minterms or product of maxterms the function is said to be in canonical form.
Example: F = x + y’z → canonical form? No
But we can put it in canonical form.
F = x + y’z = Σ (7, 6, 5, 4, 1)
Alternative way:
Obtain the truth table first and then the canonical term.

22. Standard Forms Fact: Alternative representation:
Canonical forms are very seldom the ones with the least number of literals
Alternative representation:
Standard form
a term may contain any number of literals
Two types
the sum of products
the product of sums
Examples:
F1 = y’ + xy + x’yz’
F2 = x(y’ + z)(x’ + y + z’)

23. Standard Forms
F1 = y’ + xy + x’yz’
F2 = x(y’ + z)(x’ + y + z’) ‘ *

24. Nonstandard Forms The standard form: F3 = ABC + ABD + CD + CE F3 F3 A
F3 = AB(C+D) + C(D + E)
This hybrid form yields three- level implementation D A F3 B C E
The standard form: F3 = ABC + ABD + CD + CE A B C F3 D E

25. Integrated Circuits
Many digital Logic families of IC(Integrated Circuit) have been introduced commercially. The following are the most popular:
TTL Transistor-Transistor Logic
ECL Emitter-coupled Logic
MOS Metal-oxide semiconductor
CMOS Complementary metal-oxide
semiconductor