Post on 14-Apr-2018
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BOOLEAN ALGEBRA
Boolean Algebra is the mathematics of digitalsystems.
The Boolean Expression is a logical statement
that expresses the logical relation between theinputs and the output of a digital circuit. The
output of a digital circuit is also called the
function F of the circuit.
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Boolean ExpressionsThe dot ( . ) represent AND function (also called Boolean Multiplication
).
Ex: F=A.B ( a 2-input AND)F=A.B.C.D ( a 4-input AND)
The plus ( + ) represent OR function( also called Boolean Addition).
Ex: F=A + B ( a 2-input OR)F=A+B+C ( a 3-input OR)
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Boolean ExpressionsThe bar ( ) over a letter represent the NOT function and any
inversion.
Ex: F= A ( a NOT function)
Ex: F= A.B ( a 2-input NAND)
Ex: F= A+B+C ( a 3-input NOR)
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Boolean Logic Implementation
F= E.( AB + CD)
Write the Boolean Expression for the following logic circuit:
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Combinational Logic CircuitsA combinational logic circuit employs the use of two
or more basic logic gates to form a more useful
complex function. The output of a combinational
logic circuit depends only on the current logic states
of the inputs, i.e. there isn't any sort of memory in
combinational logic circuits
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Boolean LawsNOT LAWS OR LAWS OF
COMPLEMENTATION:
LAW 1 0 = 1LAW 2 1 = 0
LAW 3 If A = 0 , then A = 1
LAW 4 If A = 1 , then A = 0
LAW 5 A ( A double bar) = A
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Boolean Laws Contd.,AND Laws
Law 6 A . 0 = 0
Law 7 A . 1 = A
Law 8 A . A = A
Law 9 A . A = 0
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Boolean Laws Contd.,OR Laws :
Law 10 A + 0 = ALaw 11 A + 1 = 1
Law 12 A + A = A
Law 13 A + A =1
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Boolean Laws Contd.,COMMUTATIVE LAWS
Law 15 A. B = B.A
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Boolean Laws Contd.,LAW 19 A +(B . C ) = (A + B) . ( A + C)
Proof :
A + BC = A + BC= A . 1 + BC LAW 7
= A ( 1 + B ) + BC LAW 11 AND LAW 14
= A + AB + BC LAW 18
= A( 1 + C) + AB + BC LAW 11= AA + AC + AB + BC LAW 8 AND 18
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Boolean Laws Contd.,A(A+C) + BA + BC LAW 18 AND 15
= A (A + C) + B ( A + C) LAW 18
= (A + C) A + (A + C) BLAW 15
= (A + C) ( A + B) LAW 18
= (A + B ) (A + C) LAW 15
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Simplification Using Boolean
LawsSimplification means reducing the number of
gates used in the circuit without changing the
function of the circuit. This will reduce cost andimprove reliability.
Examples :
Simplify the following expressions using Boolean
laws and implement using basic gates.
1. A + A B + A B
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Simplification Using Boolean
LawsSolution :Step 1 : A + A .B + A B Given :
Step 2 : A + B + AB By Law 20
Step 3 : A ( 1 + B ) + B
Step 4 : A + B By Law 11
Ans :Therefore : A + A . B + A B = A + B
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Simplification Using Boolean
Laws Contd.,Example 2 :
A B C + A B C
Solution :Given that A BC + ABC
Step 1 : B C ( A + A )
Step 2 : B C Law 13Ans : A B C + A B C = B C
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Simplification using Boolean
LawsSolution :
Given is : B ( A + B ) ( B + C )Step 1 : B ( AB + AC + BB + C )
LAW 9 B.B = 0Step 2 : B ( AB + AC + BC )Step 3 : ( ABB + ABC + B B C ) LAW 9 B.B = 0Step 4 : ( AB + ABC) LAW B.B = BStep 5 : AB ( 1 + C) LAW 11 1 + A = 1Step 6 : AB
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DeMorgans Theorems
LAW 21 A + B = A . B
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DeMorgans Law 1
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DeMorgans Law
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DeMorgans Law Contd.,
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DeMorgans Theorem contd.,
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DeMorgans Theorem contd.,
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Sum Of Products (Karnaugh
Map K Map)
0 1
1 1
A
B
A + B
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SOP (Karnaugh Map K Map)
Contd.,
3 variable Mapping:
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SOP (Karnaugh Map K Map)
Contd.,
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SOP (Karnaugh Map K Map)
Contd.,