Boolean Algebra and Logic Gates: An Introduction

Boolean Algebra
and
Logic Gate
Prof. K ADISESHA (Ph. D)

Introduction
to
Boolean Algebra
Introduction
Logical Operators
Logic Gates
Basic Theorems
Boolean Expression
2
Boolean Algebra
Prof. K. Adisesha (Ph. D)
K-Maps

Introduction
3
Introduction to Boolean Algebra:
An algebra that deals with binary number system is called “Boolean Algebra”.
➢ It is very power in designing logic circuits used by the processor of computer system.
➢ The logic gates are the building blocks of all the circuit in a computer.
➢ Boolean algebra deals with truth table TRUE and FALSE.
➢ If result of any logical statement or expression is always TRUE or 1, it is called
Tautology and if the result is always FALSE or 0, it is called Fallacy
➢ It is also called as “Switching Algebra”.

Introduction
4
History of Boolean Algebra:
Father of “Boolean Algebra”.
➢ Boolean algebra derives its name from the mathematician
George Boole (1815-1864) who is considered the
“Father of symbolic logic”.
➢ He came up with a type of Boolean algebra, the three most
basic operations of which were (and still are) AND, OR and NOT.
➢ It was these three functions that formed the basis of his premise, and were the only
operations necessary to perform comparisons or basic mathematical functions.
George Boole (1815 - 1864)

Introduction
5
Introduction to Boolean Algebra
A variable used in Boolean algebra or Boolean equation can have only one of two variables.
The two values are FALSE (0) and TRUE (1)
➢ A Sentence which can be determined to be TRUE or FALSE are called logical statements or
truth functions and the results TRUE or FALSE is called Truth values.
➢ Boolean Expression consists of
❖ Literal: A variable or its complement
❖ Product term: literals connected by •
❖ Sum term: literals connected by +
➢ A truth table is a mathematical table used in logic to computer functional values of logical
expressions.

Boolean Algebra
6
Truth Table
A variable used in Boolean algebra or Boolean equation can have only one of two variables.
The two values are FALSE (0) and TRUE (1)
➢ A truth table is a mathematical table used in logic to computer functional values of logical
expressions.
➢ A truth table is a table whose columns are statements and whose rows are possible scenarios.
➢ Example: Consider the logical expression
❖ Logical Statement: Sports = “Sunny can Play Cricket OR Football”
❖ Y = A OR B (Logical Variables: Y, A, B, Logical Operator OR)
A=Cricket B=Football Y=A OR B
0 0 0
0 1 1
1 0 1
1 1 1

Boolean Algebra
7
Logical Operators
There are three logical operator, AND, OR and NOT.
➢ These operators are now used in computer construction known as switching circuits.
➢ B = {0, 1} and two binary operators, ‘+’ and ‘.’
➢ The rules of operations: AND, OR and NOT.
❖ Complement: ~X (opposite of X)
❖ AND : X × Y
❖ OR : X + Y

Logical Operators
8
AND Operator
The operation performed by AND operator is called logical multiplication.
➢ The AND operator is read as “If and Only If”. This operator operates on two or more variables.
➢ The symbol we use for it is ‘.’
➢ Example: X . Y can be read as X AND Y
➢ The Truth table, Venn diagram and the Circuit diagram for the AND operator is.

Logical Operators
9
OR Operator
The operation performed by OR operator is called logical addition.
➢ The OR operator is read as “If at-least One”. This operator operates on two or more variables.
➢ The symbol we use for it is ‘+’.
➢ Example: X + Y can be read as X OR Y
➢ The Truth table, Venn diagram and the Circuit diagram for the OR operator is.

Logical Operators
NOT Operator
The operation performed by Not operator is called complementation.
➢ The Not operator is a unary operator. This operator operates on single variable.
➢ The symbol we use for it is bar.
❖ ~𝐗 means complementation of X
❖ If X=1 then ~X =0 If X=0 then, ~X =1
➢ The Truth table, Venn diagram and the Circuit diagram for the NOT operator is.
10

Boolean Theorems
Basic Boolean Theorems :
11

Boolean Theorems
Basic Postulates and Theorems :
12

Boolean Theorems
Principle of duality
The principle of duality states that every algebraic expression deducible from the postulates of
Boolean algebra, remains valid if the operators identity elements are interchanged:
➢ To form the dual of an expression, replace all + operators with . operators, all . operators with
+ operators, all ones with zeros, and all zeros with ones.
➢ Form the dual of the expression
❖ (A+1)=(A.0)
❖ a + (b.c) = (a + b).(a + c)
➢ Following the replacement rules:
❖ a(b + c) = a.b + a.c
➢ Take care not to alter the location of the parentheses if they are present.
13

Boolean Theorems
Idempotences Law:
Idempotences Law states that when a variable is combines with itself using OR or AND
operator, the output is the same variable”.
14

Boolean Theorems
Identity Law:
Identity Law states that when a variable is combines with 0 or 1 itself using OR or AND
operator, the output is the 0 or 1”.
15

Boolean Theorems
Absorption Law:
“This law enables a reduction of complicated expression to a simpler one by absorbing
common terms”.
16

Boolean Theorems
De-Morgan’s Theorem:
Stated by De-Morgan has two theorems:
➢ Theorem 5(a): “When the OR sum of two variables is inverted, this is same as inverting each
variable individually and then AND ing these inverted variables”
❖ (x + y)’ = x’y’
➢ Theorem 5(b): “When the AND product of two variables is inverted, this is same as
inverting each variable individually and then OR ing these inverted variables”
❖ (xy)’ = x’ + y’
➢ Can be proved by means of:
❖ Truth table or Algebraically
17

Boolean Theorems
Stated by De-Morgan has two theorems:
➢ Proof by means of truth table:
❖ (x + y)’ = x’y’ (xy)’ = x’ + y’
18
x y x’ y’ x+y (x+y)’ x’y’ xy x’+y' (xy)’
0 0 1 1 0 1 1 0 1 1
0 1 1 0 1 0 0 0 1 1
1 0 0 1 1 0 0 0 1 1
1 1 0 0 1 0 0 1 0 0

Boolean Theorems
➢ Proof by means of Algebraically:
❖ (x + y)’ = x’y’
19

Boolean Theorems
➢ Proof by means of Algebraically:
❖ (xy)’ = x’ + y’
20

Boolean Expression
Boolean Functions:
➢ A Boolean function consists of various constraints:
❖ Binary variables
❖ Binary operators OR and AND
❖ Unary operator NOT
❖ Parentheses
➢ Examples
❖ F1= x y z'
❖ F2 = x + y'z
❖ F3 = x' y' z + x' y z + x y'
❖ F4 = x y' + x' z
21

Boolean Expression
Simplification of Boolean Expression:
Simplification of Boolean expression can be achieved by two popular methods:
➢ Algebraic Manipulation
➢ Karnaugh Maps (K Map)
22

Boolean Expression
Operator Precedence
The operator precedence for evaluating Boolean Expression is:
➢ Parentheses
➢ NOT
➢ AND
➢ OR
➢ Examples
❖ x z’ + y
❖ (x y + z)'
23

Boolean Expression
Evaluation of Boolean Expression using Truth Table
To create a truth table, follow the steps given below.
➢ Step 1: Determine the number of variables, for n variables create a table with 2n rows.
❖ For two variables i.e. X, Y then truth table will need 2^2 or 4 rows.
❖ For three variables i.e. X, Y, Z, then truth table will need 2^3 or 8 rows.
➢ Step 2: List the variables and every combination of 1 (TRUE) and 0 (FALSE) for the given
variables
➢ Step 3: Create a new column for each term of the statement or argument.
➢ Step 4: If two statements have the same truth values, then they are equivalent.
24

Boolean Expression
Evaluation of Boolean Expression
Consider the following Boolean Expression F=X+Y.
➢ Step 1: This expression as two variables X and Y, then 2^2 or 4 rows.
➢ Step 2: List the variables and every combination of X and Y.
➢ Step 3: The final column contain the values of F=X+ Y.
25

Boolean Expression
Evaluation of Boolean Expression
Consider the following Boolean Expression:
➢ The truth table for the Boolean function:
is shown at the right.
➢ To make evaluation of the Boolean function easier,
the truth table contains extra (shaded) columns to hold
evaluations of subparts of the function.
26

Boolean Expression
Simplification of Boolean Expression:
Algebraic Manipulation:
➢ To minimize Boolean expressions
❖ Literal: single variable in a term (complemented or uncomplemented ) (an input to a gate)
❖ Term: an implementation with a gate
❖ The minimization of the number of literals and the number of terms → a circuit with less
equipment
❖ It is a hard problem (no specific rules to follow)
➢ Example 2.1
❖ x(x'+y) = xx' + xy = 0+xy = xy
❖ x+x'y = (x+x')(x+y) = 1 (x+y) = x+y
27

Boolean Expression
Simplification of Boolean Expression by Algebraic Manipulation:
➢ Example:
28

Boolean Expression
➢ Example:
29

Boolean Expression
➢ Example:
30

Boolean Expression
Canonical and Standard Forms :
The two canonical forms of Boolean algebra are basic forms that one obtains from
reading a given function from the truth table.
➢ We do not use it, because each minterm or maxterm must contain, by definition, all
the variables, either complemented or uncomplemented.
➢ Standard forms: the terms that form the function may obtain one, two, or any number
of literals.
❖ Sum of products: F1 = y' + xy+ x'yz'
❖ Product of sums: F2 = x(y'+z)(x'+y+z')
31

Boolean Expression
Minterms and Maxterms
➢ Boolean expression expressed as sum of Minterms or product of Maxterms are called
canonical forms.
❖ The Minterm canonical expression is the Sum of all products (SOP)
❖ The maxterm canonical expression is the product of all Sum terms (POS).
➢ For example, the following expressions are the Minterm canonical form and Maxterm
canonical form of two variables X and Y.
❖ Minterm Canonical = f(X, Y) = X’Y’ + X’ Y +X Y’+ X Y
❖ Maxterm Canonical = f(X, Y) = (X+Y).(X +Y’).(X’+Y’)
32

Boolean Expression
Minterms and Maxterms
➢ The minterms that produce a (0)
❖ f1' = m0 + m2 +m3 + m5 + m6 = x'y'z'+x'yz'+x'yz+xy'z+xyz'
❖ f1 = (f1')' = m’0 . m’2 . m’3 . m’5 . m’6 (Complement of minterms)
➢ The maxterms that produce a (1)
❖ = (x+y+z)(x+y'+z) (x+y'+z') (x'+y+z')(x'+y'+z) = M0, M2, M3, M5, M6
❖ f2 = (x+y+z)(x+y+z')(x+y'+z)(x'+y+z)=M0, M1, M2, M4
➢ Any Boolean function can be expressed as
❖ A sum of minterms (“sum” meaning the ORing of terms).
❖ A product of maxterms (“product” meaning the ANDing of terms).
❖ Both Boolean functions are said to be in Canonical form.
33

Boolean Expression
Minterms and Maxterms:
Each maxterm is the complement of its corresponding minterm, and vice versa
34

Boolean Expression
Minterms and Maxterms for three variables:
35

Boolean Expression
Minterms and Maxterms:
Any Boolean function can be expressed by:
➢ A truth table
➢ Sum of minterms
➢ f1 = x'y'z + xy'z' + xyz = m1 + m4 +m7
= S(1, 4, 7) (Minterms)
➢ f2 = x'yz+ xy'z + xyz'+xyz = m3 + m5 +m6 + m7
=P(3,5,6,7) (Minterms)
36

Boolean Expression
Minterms or Sum of Products (SOP):
The Minterm canonical expression is the Sum of all products (SOP)
➢ Sum of minterms: there are 2n minterms
➢ Example 4: Express F = A+BC’
as a sum of minterms.
➢ F(A, B, C) = Σ(1, 4, 5, 6, 7)
37

Boolean Expression
Maxterms or Product of Sum (POS):
The maxterm canonical expression is the product of all Sum terms (POS)
➢ Product of Sum terms : there are 2n minterms
➢ Example : F = xy + x'z as a product of maxterms.
➢ F(x, y, z) = Π(0, 2, 4, 5)
38

Boolean Expression
Conversion between Canonical Forms:
The complement of a function expressed as the sum of minterms equals the sum of minterms
missing from the original function.
➢ Minterm Canonical
❖ F(A, B, C) = Σ(1, 4, 5, 6, 7)
❖ Thus, F'(A, B, C) = Σ(0, 2, 3) = m0 + m2 +m3
➢ By De-Morgan's theorem
➢ Maxterm Canonical
❖ F(A, B, C) = (m0 + m2 +m3)' = M0 M2 M3 = Π(0, 2, 3)
❖ F'(A, B, C) =Π (1, 4, 5, 6, 7)
➢ Interchange the symbols Σ of 1’s and Π of 0’s and list those numbers missing from the original form.
39

Boolean Expression
Conversion between Canonical Forms:
Two different ways to specify the same function f of three variables.
➢ Minterm Canonical
❖ SOP Form
❖ f(x,y,z) = Σ m(1, 3, 6, 7)
➢ Maxterm Canonical
❖ POS Form
❖ f(x,y,z) = Π M(0, 2, 4, 5)
40

Boolean Expression
Conversion of SOP into Canonical form:
Convert the Boolean function f(X, Y) = X + X Y into canonical form.
➢ Solution:
❖ The given Boolean function f(X, Y) = X + X Y ------ (i)
❖ It has two variables and sum of two Minterms. The first term X is missing one variable.
❖ So to make it of two variables it can be multiplied by (Y+Y’)=1.
❖ Therefore, X = X (Y+Y’) =XY+XY’
❖ Substitute the value of X in (i) we get f(X, Y) = X Y+X Y’ + X Y
❖ Here, the term X Y appear twice, it is possible to remove one of them. f(X, Y) = X Y+X Y’
❖ Therefore: SOP Expression is f(X, Y) = Σ (2, 3)
41

Boolean Expression
Conversion of POS into Canonical form:
Convert the Boolean function F(X, Y, Z) = X+Y (Y+Z) into canonical form.
➢ Solution:
❖ The given Boolean function F(X,Y,Z) = (X+Y).(Y+Z) ------(i)
❖ It has three variables and product of two Maxterms. Each Maxterm is missing one variable.
❖ The first term can be written as X+Y = (X+Y+Z. Z ) Since Z. Z =0
❖ Using distributive law (X + YZ) = (X + Y) (X + Z), we can write X+Y = (X+Y+Z) (X+Y+ Z ) ---(ii)
❖ The Second term can be written as Y + Z = (Y+Z+X. X ) Y+Z = (Y+Z+X) (Y+Z+X ) ----(iii)
❖ Substitute (ii) and (iii) in (i) we get
F(X, Y, Z) = (X+Y+Z) (X+Y+ Z ) (Y+Z+X) (Y+Z+X )
❖ Therefore: POS Expression is F(X, Y, Z) = π (0, 1, 4)
42

Karnaugh Map
Karnaugh Map:
A graphical display of the fundamental products in a truth table.
➢ Fundamental Product: The logical product of variables and complements that produces a high
output for a given input condition.
➢ The map method provides simple procedure for minimizing the Boolean function.
➢ The map method was first proposed by E.W. Veitch in 1952 known as “Veitch Diagram”.
➢ In 1953, Maurice Karnaugh proposed “Karnaugh Map” also known as “K-Map”.
43

Karnaugh Map
Construction of K-Map :
The K-Map is a pictorial representation of a truth table made up of squares.
➢ Each square represents a Minterm or Maxterm.
➢ The Karnaugh map is completed by entering a '1‘(or ‘0’) in each of the appropriate cells.
➢ A K-Map for n variables is made up of 2n squares.
❖ Single Variable K-Map: The map consists of 2 squares (i.e. 2n square, 21
= 2 square)
❖ Two Variable K-Map : The map consists of 4 squares (i.e. 2n
square, 22
= 4 square)
❖ Three Variable K-Map: The map consists of 8 squares (i.e. 2n square, 23
= 8 square)
❖ Four Variable K-Map : The map consists of 16 squares (i.e. 2n
square, 24
= 16 square)
44

Karnaugh Map
Karnaugh maps, or K-maps, are often used to simplify logic problems with 2, 3 or 4
variables.
➢ Within the map, adjacent cells containing 1's (or 0’s) are grouped together in twos, fours, or
eights.
➢ For the case of 2 variables, we form a map consisting of 22
=4 cells
➢ as shown in Figure .
45

Karnaugh Map
Karnaugh maps, or K-maps, are often used to simplify logic problems with 3 or 4
variables.
46
AB
C 00 01 11 10
0
1
CBA CBA CAB CBA
CBA BCA ABC CBA
0 2 6 4
531 7
3 Variables Cell = 23=8

Digital Logic Gates
47
Logic Gates:
A logic gate is an idealized model of computation or physical electronic device
implementing a Boolean function.
➢ A logical operation performed on one or more
binary inputs that produces a single binary output.
➢ Types of Logic Gates:
➢ Basic Gates
➢ Universal Gates
➢ Exclusive Gates

Types of Logic Gates:
Basic Gates:
Digital Logic Gates
48

Universal Gates:
Digital Logic Gates
49

Exclusive Gates:
Digital Logic Gates
50

NAND Gate is a Universal Gate
51
To prove that any Boolean function can be implemented using only NAND gates, we will show that the
AND, OR, and NOT operations can be performed using only these gates

NOR Gate is a Universal Gate
52
To prove that any Boolean function can be implemented using only NOR gates, we will show that the
AND, OR, and NOT operations can be performed using only these gates

Realization of Basic Gates using Universal Gates:
NAND Gates:
Digital Logic Gates
53

Realization of Basic Gates using Universal Gates:
NOR Gates:
Digital Logic Gates
54

Realization of Exclusive Gates using Universal Gates:
Using NAND Gates: Using NOR Gates:
Digital Logic Gates
55

Binary Adder:
In many computers and other kinds of processors adders are used in the
arithmetic logic units or ALU.
➢ An adder is a device that will add together two bits and give the result as the
output.
➢ There are two kinds of adders –
❖ Half adders: A half adder just adds two bits together and gives a two-bit
output
❖ Full adders: A full adder adds two inputs and a carried input from another
adder, and also gives a two-bit output.
Digital Logic Gates
56

Realization of Half Adder Circuits using Logic Gates:
Half adders: A half adder just adds two bits together and gives a two-bit output
Digital Logic Gates
57

Realization of Full Adder Circuits using Logic Gates:
Full adders: A full adder adds two inputs and a carried input from another adder,
and also gives a two-bit output.
Digital Logic Gates
58

Digital Logic Gates
59
Integrated Circuits:
A collection of one or more gates fabricated on a single silicon chip is called an
integrated circuit (IC).
➢ ICs were classified by size:
❖ SSI - small scale integration - 1~20 gates
❖ MSI - medium scale integration - 20~200 gates
❖ LSI - large scale integration - 200~200,000 gates
❖ VLSI - very large scale integration - over 1M transistors
➢ Pentium-III - 40 million transistors

Digital Logic Gates
60
Integrated Circuits:

Digital Logic Gates
61
DIP Packages

Digital Logic Gates
62
Gates in ICs

NOR Gate is a Universal Gate
63
Implementation with logic gates

Boolean Functions
64
Implementation with logic gates

Boolean Functions
65
Multiple Inputs Gates
Multiple input NOR = a complement of OR gate, Multiple input NAND = a complement of AND.
➢ The cascaded NAND operations = sum of products.
➢ The cascaded NOR operations = product of sums.

Boolean Functions
66
Multiple Inputs Gates
The XOR and XNOR gates are commutative and associative.
➢ XOR is an odd function: it is equal to 1 if the inputs variables have an odd number of 1's.

Boolean Functions
67
Positive and Negative Logic
The XOR and XNOR gates are commutative and associative.
➢ Positive and Negative Logic
❖ Two signal values <=> two logic values
❖ Positive logic: H=1; L=0
❖ Negative logic: H=0; L=1

Discussion
68
Queries ?
Prof. K. Adisesha
9449081542

Boolean Algebra and Logic Gates: An Introduction

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Boolean Algebra and Logic Gates: An Introduction