Unit 1
BOOLEAN ALGEBRA
INTRODUCTION
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.
Tautology & 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’
Implementation
Boolean Algebra applied in
computers electronic circuits. These
circuits perform Boolean operations
and these are called logic circuits or
logic gates.
Logic Gate
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
AND gate
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 Input Output
A B AB
0 0 0
0 1 0
1 0 0
1 1 1
OR gate
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 Input Output
A B A+B
0 0 0
0 1 1
1 0 1
1 1 1
NOT gate
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
PRACTICAL APPLICATIONS OF LOGIC GATES
PRACTICAL APPLICATIONS OF LOGIC GATES
AND Gate
So while going out of the house you set
the "Alarm Switch" and if the burglar enters he
will set the "Person switch", and tada the alarm
will ring.
PRACTICAL APPLICATIONS OF LOGIC GATES
AND Gate
Electronic door will only open if it
detects a person and the switch is set to
unlocked.
Microwave will only start if the start
button is pressed and the door close
switch is closed.
PRACTICAL APPLICATIONS OF LOGIC GATES
OR Gate
You would of course want your doorbell
to ring when someone presses either the front
door switch or the back door switch..(nice)
PRACTICAL APPLICATIONS OF LOGIC GATES
NOT Gate
When the temperature falls below 20c
the Not gate will set on the central heating
system (cool huh).
NAND, NOR XOR, XNOR GATES
NAND Gate
Known as a “universal” gate
because ANY digital circuit can be
implemented with NAND gates
alone.
NAND Gate
NAND
X Y Z
X 0 0 1
Z 0 1 1
Y 1 0 1
1 1 0
Z = ~(X & Y)
nand(Z,X,Y)
NAND Gate
F = (X•X)’ X F = X’
X
= X’+X’
= X’
X X F X•Y
F = ((X•Y)’)’
Y Y
= (X’+Y’)’
= X’’•Y’’
= X•Y
X X
F = (X’•Y’)’ F = X+Y
= X’’+Y’’
Y Y
= X+Y
NOR Gate
NOR Gate
NOR
X Y Z
X 0 0 1
Z
Y 0 1 0
1 0 0
Z = ~(X | Y)
1 1 0
nor(Z,X,Y)
Exclusive-OR Gate
Exclusive-OR Gate
XOR XY Z
X Z
Y 00 0
Z=X^Y 01 1
10 1
xor(Z,X,Y) 11 0
Exclusive-NOR Gate
Exclusive-NOR Gate
XNOR XY Z
X Z
Y 00 1
01 0
Z = ~(X ^ Y) 10 0
Z = X ~^ Y 11 1
xnor(Z,X,Y)
POWER CONSUMPTION OF SYSTEM
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
Basic Theorem of Boolean Algebra
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
Basic Theorem of Boolean Algebra
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
Basic Theorem of Boolean Algebra
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
De Morgan's Theorem 1
Theorem 1 A . B = A + B
De Morgan's Theorem 1
Theorem 1 A . B = A + B
De Morgan's Theorem 1
Theorem 1 A . B = A + B
De Morgan's Theorem 2
Theorem 1 A + B = A . B
De Morgan's Theorem 2
Theorem 2 A + B = A . B
De Morgan's Theorem 2
Theorem 2 A + B = A . B
De Morgan's Theorem 2
Theorem 2 A + B = A . B
