Truth Table:
Relationship between a
function and variable
Logic Diagram:
Algebraic
Expression
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Purpose of Boolean Algebra
To facilitate the analysis and design of digital circuit
F= AB’ + C’D + AB’ + C’D
= x + x (let x= AB’ + C’D)
= x
= AB’ + C’D
Boolean function = Algebraic form = convenient tool
Truth table (relationship between binary variables
)
Algebraic form
Logic diagram (input-output relationship
) Algebraic form
Find simpler circuits for the same function :
by using
Boolean algebra rules
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Graphic symbols for NOR & NAND gate
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Rule in Boolean algebra
Variable used can have only two values.
Binary 1 for HIGH and Binary 0 for LOW.
Complement of a variable is represented by
an over bar (-).
Thus complement of variable B
is represented as.
Thus if B = 0 then = 1 and B = 1 then = 0.
ORing of the variables is represented by a plus
(+) sign between them.
For example ORing of A, B, C is represented as A +
B + C.
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Cont…
Logical ANDing of the two or more
variable is represented by writing a dot
between them
such as A.B.C. Sometime the dot may be
omitted like ABC.
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Boolean Laws
COMMUTATIVE LAW
A.B=B.A
OR
A+B=B+A
ASSOCIATIVE LAW
(A.B).C= A.(B.C) OR (A+B)+C= A+(B+C)
DISTRIBUTIVE LAW
A.(B+C)=(A.B)+(A.C)
AND LAW
A.0=0
A.1=1
A.A=A
A.A=O
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Cont…
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OR LAW
A.0=0
A.1=A
A.A=A
A.A=0
INVERSION LAW
The inversion law states that double inversion of
a variable result in the original variable itself.
A ’ ’=A

Truth Table Formation
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A truth table represents a table having all
combinations of inputs and their corresponding
result

Map Simplification
Karnaugh Map(K-Map)
Map method for simplifying Boolean expressions
Minterm / Maxterm
Minterm : n variables
product ( x=1, x’=0)
Maxterm : n variables
sum (x=0, x’=1)
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F(x, y, z) = Σ(1,4,5,6,7)
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Adjacent Square
Number of square = 2n (2, 4, 8, ….)
The squares at the extreme ends of the
same horizontal row are to be considered
adjacent
The same applies to the top and bottom
squares of a column
The four corner squares of a map must be
considered to be adjacent
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Adjacent Squares
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Example 1,2
F(A,B,C) = Σ(3,4,6,7)
F=AC’ + BC
F(A,B,C) = Σ(0,2,4,5,6)
F=C’ + AB’
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Example 3
F(A,B,C,D) =
Σ(0,1,2,6,8,9,10)
F=B’D’ + B’C’ +
A’CD’
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LOGIC GATE
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- Fall '19
- Boolean Algebra, Logic gate, Combinational Circuits, Examples of the digital systems