Introduction:
·
The
algebra of logic, which deals with the study of binary variables and logical
operations is called Boolean Algebra.
·
Today,
it is the backbone of design and analysis of computer and other digital
circuits.
·
They
are useful not only to the hardware design in implementing circuits, but also
to software designer in making decision and for many tasks.
·
In
Boolean algebra the variables are permitted to have two values. They are
a)
Logic
1 ( i. e. on / yes / true / high)
b)
Logic
0 ( i. e. off / no / false / low)
Boolean Variables(Logical
Variables):
The
variables that have only two values 1 and 0 are called Boolean Variables or Logical Variables. These variables are denoted
by A, B, X, Y etc.
Truth Table:
A
table which represents the input output relationships of the binary variables
for each gate is called Truth Table.
It shows the relation between all inputs outputs in tabular form. Thus, a truth
table is a table representing the results of the logical operations on all
possible combinations of logical values.
Boolean Expression
(Boolean Function)
An
expression formed by binary variables, binary operators (AND, OR, NOT) ,
parenthesis and equal sign is called Boolean Expression or Boolean Function.
For a given value of the variables, the function can be either 0 or 1.
For
Example: Z = X.Y
Logic Gate:
An
electronic circuit that operates on one or more input signals to produce an
output signal is called logic gate. The logic gate is used for binary operation
and is the basic component of digital computer. Each gate has its specific
function and graphical symbol.
There
are 3 basic gates: 1) AND gate 2) OR gate 3) NOT gate
From
the combination of these 3 basic gates, we can get other derived gates, which
are:
4)
NAND gate 5) NOR
gate 6) Exclusive – OR
(XOR) gate
7)
Excusive – NOR (XNOR) gate
1. AND
gate:
An electronic circuit
which produces high(1) output when all inputs are high(1) otherwise, produces
the output low (0) is called AND gate. The output of AND gate is equal to the
product of the logic inputs. It can have two or more inputs and produces only
one output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


2. OR
gate
An electronic circuit which
produces high (1) output when all inputs are high (1) or any one of input is
high (1) and produces output low (0) when all inputs are low (0) is called OR gate. The output of the OR gate is
equal to the sum of the logic inputs. It has two or more inputs and produces
only one output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


3. NOT
gate (Inverter)
An electronic circuit whose output is the complement
of the input is called NOT gate. It is also called an inverter. If we
provide high (1) input to NOT gate, it will produce low (0) output and
vice versa. It has only one input and only one output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


4. NAND
gate
The
gate which is formed by the combination of AND gate and NOT gate is called NAND
gate. NAND gate produces low (0) output, when all inputs are high (1);
otherwise, produces high (1) output. It is the complement of the AND gate. It
has two or more inputs and only one output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


5. NOR
gate
The gate which is formed by the combination of OR gate
and NOT gate is called NOR gate. NOR gate produces high (1) output, when all
inputs are low (0); otherwise, output will be low (0). It is the complement of
OR gate. It has two or more inputs and only one output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


6. Exclusive
– OR (XOR) gate
The
XOR gate produces low output (0) when the both inputs are same; otherwise, the
output will be high (1). It can also have two or more inputs and only one
output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


7. Exclusive
NOR (XNOR) gate
The XNOR gate is
equivalent to an XOR gate followed by an inverter. This gate produces high(1)
output when all inputs are either low(0) or high(1). It can also have two or
more inputs and a single output.
Name

Graphical
Symbol

Algebraic
Function

Truth
Table


# Duality Principle:
Dual of a
Boolean expression is derived by
1.
Replacing
AND operation by OR
2.
Replacing
OR operation by AND
3.
All
1’s are changed to 0
4.
All
0’s are changed to 1
5.
Variables
and complements are left unchanged.
For
Example:
F=(X’.1).(0+X)
, Here dual of F= (X’+0)+(1.X)
# Laws of Boolean algebra Associative,
Commutative, Distributive, Identity, Complement Law:
1. Associative
Laws:
The associative law of Boolean
algebra is expressed by:
* (A+B)+C=A+(B+C)
* (A+B)+C=A+(B+C)
*
(A.B).C=A.(B.C)
Proof for
(A+B)+C=A+(B+C)
Truth Table:
Inputs


Output 1


Output 2


A

B

C

A+B

(A+B)+C

B+C

A+(B+C)

0

0

0

0

0

0

0

0

0

1

0

1

1

1

0

1

0

1

1

1

1

0

1

1

1

1

1

1

1

0

0

1

1

0

1

1

0

1

1

1

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

1

Conclusion:
Comparing
the values of (A+B)+C and A+(B+C) from truth table, since ,both are equal.
Hence, Proved.
2. Commutative
Laws:
The
commutative law of Boolean algebra is expressed by:
* (A+B)=(B+A)
* (A.B)=(B.A)
Proof for (A+B)=(B+A)
Truth Table:
Inputs

Output 1

Output 2


A

B

A+B

(B+A)

0

0

0

0

0

1

1

1

1

0

1

1

1

1

1

1

Conclusion:
Comparing
the values of (A+B) and (B+A) from truth table, since, both are equal. Hence,
proved.
3. Distributive
Laws:
A
distributive law of Boolean algebra is expressed by:
*
A.(B+C)=(A.B)+(A.C)
*
A+(B.C)=(A+B).(A+C)
Proof for a. (B+C)=(A.B)+(A.C)
Truth
Table:


Output
1


Output
2


A

B

C

B+C

A.(B+C)

A.B

A.C

(A.B)+(A.C)

0

0

0

0

0

0

0

0

0

0

1

1

0

0

0

0

0

1

0

1

0

0

0

0

0

1

1

1

0

0

0

0

1

0

0

0

0

0

0

0

1

0

1

1

1

0

1

1

1

1

0

1

1

1

0

1

1

1

1

1

1

1

1

1

Conclusion:
Comparing
the values of A.(B+C) and (A.B)+(A.C) from truth table, since ,both are equal.
Hence, Proved.
1. Identity
Laws:
The
identify law of Boolean algebra is expressed by:
*
A+0 = A and A.1=A
Proof for A+0=A
Truth
Table
Inputs

Output


A

0

A+0

0

0

0

1

0

1

Conclusion:
Hence from
the truth table the output will be same as the input.
2. Complement
Laws:
The
complement law of Boolean algebra is expressed by:
*
A+A’=1 and A.A’=0
Proof for A+A’=1
Truth Table:
Inputs

Output


A

A’

A+A’

0

1

1

1

0

1

Conclusion:
Hence from
the truth table the sum of an input and its complement will always be true (1).
Universal gate (NAND and NOR gate):
NAND Gate:
NOR gate:
#
De Morgan’s Theorem:
First
Theorem:
The De Morgan’s first theorem states that “The
complement of a sum equals to the product of the complements of individual”.
i.e. (A+B)’=A’.B’
Truth
Table:
Inputs


Output
1



Output
2


A

B

(A+B)

(A+B)’

A’

B’

A’.B’

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

Conclusion:
Comparing the values of (A+B)’ and A’.B’ from the
truth table, both are equal, hence, proved.
Second Theorem:
The
De Morgan’s first theorem states that “The complement of a product is equal to the sum of the complements of
individual”. i.e. (A.B)’=A’+B’
Truth
Table:
Inputs


Output
1



Output
2


A

B

(A.B)

(A.B)’

A’

B’

A’+B’

0

0

0

1

1

1

1

0

1

0

1

1

0

1

1

0

0

1

0

1

1

1

1

1

0

0

0

0

Conclusion:
Comparing
the values of (A.B)’ and A’+B’ from the truth table, both are equal, hence,
proved.
#
Venn Diagram:
Venn
diagram is the diagram in which areas represents operations of logic gates.
Homework [Unit4]
1.
Describe the De Morgan’s Law.
2.
Define Boolean algebra. Explain AND, OR ,
NAND and XOR gate with truth table and logic gate.
3.
Describe any four logic gates with truth
table and gate symbol.
4.
Differentiate between OR and AND gate.
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