UNIT-4 Logic Function and Boolean
Algebra
4.1.Logic Function and Boolean Algebra
4.2.Introduction of Truth Table, Boolean Expression
4.3.Logic Gates –AND, OR, NOT, NAND, NOR, XOR and XNOR
– its definition, use, truth table, logic symbol.
4.4. Duality Principle
4.5.Laws of Boolean algebra –
Associative, Commutative, Distributive, Identity, Complement Laws.
4.6.De Morgan’s Theorem : Statement and Logic Expression
4.7.Venn diagram and its represent of logic gates(AND,
OR, NOT)
|
Logic Function:- A logic function is an
expression expressed algebraically with binary variables, logical operation
symbols, parenthesis and equal sign, is also known as Boolean function. For
example, in the logic function F=A+B, the value of F is 0 if A=0 and B=0
otherwise the value of F is 1.The logical function can be represented as logic
diagram composed of AND, OR and NOT gates.
Boolean algebra:
Boolean
algebra is the branch of mathematics that includes methods for manipulating
logical variables and logical expression. An algebra in which all elements can take only one of
two values (typically 0 and 1, or
"true" and "false") and are subject to operations based on AND, OR and NOT is called Boolean algebra. Boolean algebra is
the algebra of two values 0 and 1.
The
name “Boolean” comes from the name of a logician called George Boole, an
English mathematician born in 1815.
About George Boole:
Born
on 2nd November, 1815 in industrial town, Lincoln, of England.
Famous
mathematician and logician.
Founder
of Boolean algebra
Truth Table:
A truth table is the tabular representation of Boolean
function used in logic to compute the functional value of logical expression on
each of their functional arguments. In other word, A truth Table is defined as
a table which represents the input and output relationship of the binary
variables for each gate. For example:
Inputs
|
Output
|
|
A
|
B
|
Y=A.B
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
1
|
1
|
Boolean expression:-
A Boolean expression is a string of symbols
representing logical variable and logical operations which is evaluated to give
a logical value. For example: A+AB, AB+AB’ etc.
Logic gate:
A logic gate is an
electronic circuit which generates only one output signal from one or more
input signals.l. In other word, the basic elements of circuit are called gates.
Each type of gate implemented Boolean operation. The gates are AND, OR, NAND,
NOR, NOT, XOR, XNOR.
a)
AND Gate:
Definition:
The AND gate is a logic
device that has two or more inputs and one output. The AND gate produces a
logic 1 at its output only when all of the inputs are high. A logic low state
is produced when one or more of the inputs are low.
Boolean Equation:
X = A B C
The symbol for an AND
operation is a center dot ().
Truth
Table:
Input
|
Output
|
||
A
|
B
|
C
|
X
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
Symbol:
b)
OR Gate:
Definition:
The OR gate is a logic
device that has two or more inputs and one output.
The OR gate produces a
logic 1 when one or more of its inputs are high. A logic low state is produced
only when all of the inputs are low.
Boolean Equation:
X = A + B + C
The symbol for an OR operation
is a +.
Truth
Table:
Input
|
Output
|
||
A
|
B
|
C
|
X
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
Symbol:
c)
NOT Gate (Inverter):
Definition:
The inverter is a logic
device that only has one input and one output.
The inverter performs
the logic function called inversion or complementation. If the input signal is
low then high output signal is obtained and vice versa.
Boolean Equation:
A complement bar (also
called an overbar) is placed over the assigned input letter.
The expression is reads
as "X" is equal to NOT "A".
Truth
Table:
Input
|
Output
|
A
|
X
|
1
|
0
|
0
|
1
|
Symbol:
d)
NAND Gate:
Definition:
The NAND gate is a
logic device that has two or more inputs and one output.
The NAND gate produces
a logic 0 at its output only when all of the inputs are logic 1. A logic 1 is
produced when any of the inputs are low.
Boolean Equation:
Truth
Table:
Input
|
Output
|
||
A
|
B
|
C
|
X
|
1
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
1
|
Symbol:
e)
NOR Gate:
Definition:
The NOR gate is a logic
device that has two or more inputs and one output.
The NOR gate produces a
logic 1 at its output only when all inputs are "low". A logic 0 is generated if any of the inputs is
a "high".
Boolean Equation:
Truth
Table:
Input
|
Output
|
||
A
|
B
|
C
|
X
|
1
|
1
|
1
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
1
|
Symbol:
f)
XOR Gate:
Definition:
The Exclusive - OR (or Ex-OR)
gate is a logic device that always has two inputs and one output.
The XOR gate produces a
logic 1 at its output only when the inputs are at opposite states. A logic 0 is
produced when the two inputs are the same.
Truth
Table:
Input
|
Output
|
|
A
|
B
|
X
|
1
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
1
|
0
|
0
|
0
|
Symbol:
g)
X-NOR Gate:
Definition:
The Exclusive - NOR (
or Ex-NOR) gate is a logic device that always has two inputs and one output.
The X-NOR gate produces
a logic 1 at its output only when the inptus are at same states. A logic 0 is
produced when both inputs are of opposite states.
Boolean Equation:
Truth
Table:
Input
|
Output
|
|
A
|
B
|
X
|
1
|
1
|
1
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
Symbol:
Universal gate
A universal gate is a gate which can implement
any Boolean function without using any other types of gates. NAND and NOR gates
are known as universal gates.
Why
NAND or NOR gate is called universal gate?
NAND AND NOR gates are called universal gates , because with
a combination NAND and NOR gates alone, it's possible to create all other logic
gates like AND, OR, XOR etc and you can design any logic circuit. In other
word, NAND and NOR Gates are called Universal Gates because all the other
gates can be created by using these gates
Principle of Duality
According to principle of Duality, dual of a Boolean
expression can be obtained by replacing AND (.) with OR (+) and vice versa, 1
with 0 and vice versa keeping variables and complements and variables are
unchanged. For example, duality of the expression (A+0) is (A.1), A.B’ +C is
A+B’.C
Laws of Boolean Algebra
a. Associative
Law
The associative law states that there is no effect of ORing
or ANDing operation on the method of variable grouping.
i.
(A+B) + C = A +
(B+C)
ii.
(A.B).C = A.(B.C)
Proof:
b. Distributive
Law
i.
A.(B+C) = A.B +
A.C
Proof:
If A=1, B=0,
C=1 then
LHS = 1.(0+1)=
1.1 = 1
RHS = 1.0+1.1=
0 + 1 = 1
\ LHS = RHS
c. Commutative
Law
The commutative law states that the order in which you
add or multiply input signals does not affect the result.
i.
A+B = B+A
ii.
A.B = B.A
Proof:
Principle of Duality
The dual of any statement in a Boolean
algebra is the statement obtained by interchanging +(OR) and .(AN
Augustus De Morgan
- Born in Jan. 27, 1806 and died in March 18, 1871
- He was entered in
- He is famous for his theorems known as De Morgan
theorems which is a foundation of many logic circuits used in electronic
equipments.
De Morgan’s Theorem
First Theorem:
The complement of a sum equals to the
product of the complements.
(A+B)’=A’.B’
Proof:
A
|
B
|
A’
|
B’
|
A+B
|
(A+B) ’
|
A’.B’
|
0
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
0
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
0
|
Second Theorem:
The complement of a product equals the
sum of the complements.
(A.B)’=A’+B’
Proof:
A
|
B
|
A’
|
B’
|
A.B
|
(A.B) ’
|
A’+B’
|
0
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
0
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
1
|
1
|
1
|
1
|
0
|
0
|
1
|
0
|
0
|
Labels: CH-4
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