Chapter 3 Boolean Algebra_XI_B_boolean logic circuits

CHAPTER – 03
BOOLEAN ALGEBRA

INTRODUCTION
Developed by English Mathematician
George Boole in between 1815 - 1864.
It is described as an algebra of logic or
an algebra of two values i.e True or
False.
The term logic means a statement
having binary decisions i.e True/Yes or
False/No.

APPLICATION OF BOOLEAN ALGEBRA

APPLICATION OF BOOLEAN ALGEBRA
• It is used to perform the logical
operations in digital computer.
• In digital computer True represent by ‘1’
(high volt) and False represent by ‘0’ (low
volt)
• Logical operations are performed by
logical operators. The fundamental logical
operators are:
1. AND (conjunction)
2. OR (disjunction)
3. NOT (negation/complement)

AND operator
It performs logical multiplication and denoted
by (.) dot.
X Y X.Y
0 0 0
0 1 0
1 0 0
1 1 1

OR operator
It performs logical addition and denoted
by (+) plus.
X Y X+Y
0 0 0
0 1 1
1 0 1
1 1 1

NOT operator
It performs logical negation and
denoted by (-) bar. It operates on single
variable.
X X (means complement of x)
0 1
1 0

Truth Table
• Truth table is a table that contains all
possible values of logical
variables/statements in a Boolean
expression.
No. of possible combination =
2n, where n=number of variables used in
a Boolean expression.

Truth Table
The truth table for XY + Z is as follows:
Dec X Y Z XY XY+Z
0 0 0 0 0 0
1 0 0 1 0 1
2 0 1 0 0 0
3 0 1 1 0 1
4 1 0 0 0 0
5 1 0 1 0 1
6 1 1 0 1 1
7 1 1 1 1 1

Tautology & Fallacy
If the output of Boolean
expression is always True or 1 is
called Tautology.
If the output of Boolean
expression is always False or 0 is
called Fallacy.

Exercise
1. Evaluate the following Boolean
expression using Truth Table.
(a) X’Y’+X’Y (b) X’YZ’+XY’
(c) XY’(Z+YZ’)+Z’
2. Verify that P+(PQ)’ is a Tautology.
3. Verify that (X+Y)’=X’Y’

X Y X' Y' X'Y' X'Y X'Y' + X'Y
0 0 1 1 1 0 1
0 1 1 0 0 1 1
1 0 0 1 0 0 0
1 1 0 0 0 0 0
X’Y’+X’Y

X’YZ’+XY’
X Y Z X' Y' Z' X'YZ' XY'
X'YZ' +
XY'
0 0 0 1 1 1 1 0 1
0 0 1 1 1 0 0 0 0
0 1 0 1 0 1 1 0 1
0 1 1 1 0 0 0 0 0
1 0 0 0 1 1 0 1 1
1 0 1 0 1 0 0 1 1
1 1 0 0 0 1 0 0 0
1 1 1 0 0 0 0 0 0

XY’(Z+YZ’)+Z’
X Y Z Y' Z' YZ' Z + YZ' XY'
XY'(Z +
YZ')
Final:
XY'(Z +
YZ') + Z'
0 0 0 1 1 0 0 0 0 1
0 0 1 1 0 0 1 0 0 0
0 1 0 0 1 1 1 0 0 1
0 1 1 0 0 0 1 0 0 0
1 0 0 1 1 0 0 1 0 1
1 0 1 1 0 0 1 1 1 1
1 1 0 0 1 1 1 0 0 1
1 1 1 0 0 0 1 0 0 0

Verify that P+(PQ)’ is a Tautology.
P Q PQ (PQ)' P + (PQ)'
0 0 0 1 1
0 1 0 1 1
1 0 0 1 1
1 1 1 0 1

Verify that (X+Y)’=X’Y’
X Y X+Y (X+Y)' X' Y' X'Y'
0 0 0 1 1 1 1
0 1 1 0 1 0 0
1 0 1 0 0 1 0
1 1 1 0 0 0 0

Implementation
Boolean Algebra applied in
computers electronic circuits. These
circuits perform Boolean operations
and these are called logic circuits or
logic gates.

Logic Gate
Purpose: we use logic gates to allow electrical signals to
the digital electronic circuits.
The digital devices which perform Boolean functions by
taking 1 or more i/p and giving 1 o/p is called Logic
Gate.
Logic Gates are based on Boolean algebra.
Logic gates are made up of diode and transistor.

Logic Gate
A gate is an digital circuit which
operates on one or more signals and
produce single output.
Gates are digital circuits because the
input and output signals are denoted by
either 1(high voltage) or 0(low voltage).
There are three basic gates and are:
1. AND gate 2. OR gate
3. NOT gate

Chapter 3 Boolean Algebra_XI_B_boolean logic circuits

AND gate
• The AND gate is an electronic circuit that
gives a high output (1) only if all its inputs are
high.
• AND gate takes two or more input signals and
produce only one output signal.
Input
A
Input
B
Output
AB
0 0 0
0 1 0
1 0 0
1 1 1

OR gate
• The OR gate is an electronic circuit that gives
a high output (1) if one or more of its inputs
are high.
• OR gate also takes two or more input signals
and produce only one output signal.
Input
A
Input
B
Output
A+B
0 0 0
0 1 1
1 0 1
1 1 1

NOT gate
• The NOT gate is an electronic circuit that gives
a high output (1) if its input is low .
• NOT gate takes only one input signal and
produce only one output signal.
• The output of NOT gate is complement of its
input.
• It is also called inverter.
Input A Output A
0 1
1 0

NAND Gate
NAND
X
Y
Z
X Y Z
0 0 1
0 1 1
1 0 1
1 1 0

NAND Gate
X
X
F = (X•X)’
= X’+X’
= X’
X
Y
Y
F = ((X•Y)’)’
= (X’+Y’)’
= X’’•Y’’
= X•Y
F = (X’•Y’)’
= X’’+Y’’
= X+Y
X
X
F = X’
X
Y
Y
F X•Y
F = X+Y

NOR Gate
NOR
X
Y
Z
X Y Z
0 0 1
0 1 0
1 0 0
1 1 0

Exclusive-OR Gate
X Y Z
XOR
X
Y Z 0 0 0
0 1 1
1 0 1
1 1 0

Exclusive-NOR Gate
X Y Z
XNOR
X
Y Z 0 0 1
0 1 0
1 0 0
1 1 1

NAND Gate
Known as a “universal” gate
because ANY digital circuit can be
implemented with NAND gates alone.

Basic Theorem of Boolean Algebra
T1 : Properties of 0
(a) 0 + A = A
(b) 0 A = 0
T2 : Properties of 1
(a) 1 + A = 1
(b) 1 A = A

T3 : Commutative Law
(a) A + B = B + A
(b) A B = B A
T4 : Associate Law
(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)
T5 : Distributive Law
(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)
(c) A+A’B = A+B

T6 : Indempotence (Identity ) Law
(a) A + A = A
(b) A A = A
T7 : Absorption (Redundance) Law
(a) A + A B = A
(b) A (A + B) = A

T8 : Complementary Law
(a) X+X’=1
(b) X.X’=0
T9 : Involution
(a) x’’ = x
T10 : De Morgan's Theorem
(a) (X+Y)’=X’.Y’
(b) (X.Y)’=X’+Y’

De Morgan's Theorem 1
Theorem 1 A . B = A + B

Theorem 1 A + B = A . B

Theorem 2 A + B = A . B

Chapter 3 Boolean Algebra_XI_B_boolean logic circuits

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