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Boolean ExpressionSimplification
Algebra Method
Map Method
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Algebraic ManipulationThe use of identities to simplify boolean
expressionsDe Morgan’s, Associativity, etc.Awkw r in i l k ifi r l
predict each succeeding step in themanipulative process
Difficult to determine whether the simplestexpression has been achieved.
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Map MethodProvides a straightforward procedure forsimplifying Boolean functions up to fourvariables.The map is also known as the Karnaugh mapor K-map.The map is a diagram made up of squareswith each square representing one mintermof the function.
A Boolean function is recognized graphicallyin the map by those squares whose mintermsare included in the function.
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Map MethodBy recognizing various patterns, the user can
derive alternative algebraic expressions forthe same function, from which the simplestcan be selected.
map are always in sum-of-products orproduct-of-sum form.
It is sometimes possible to produce two ormore expressions that satisfy thesimplification criterion.
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Types of Maps- According to number of variables
Two-Variable MapThree-Variable Map
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Two-variable mapA’B’ A B’
A’B A Bm0 m2
m1 m3
A
0 2
1 3
Fig. (a)Fig. (b)
Fig. (c)
BA
0
0
1
1
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Two-variable mapExample : f(A,B)= Σ m(0,3)
1B
A
0
0 1What is the algebraic
11 the map?
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Two-variable mapExample
1A
B
0
0 1 What is the logical sumand the boolean
1 11
A’B’ A’ B
A B’ A B
express on representeby the map?
0 1
2 3
A B
00
1
1
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Two-variable mapSimplifying
1A
B 0 1 m1+m2+m3= A’B+AB’+AB= A’B+A(B’+B)
’1 11
= += A+B
A’B’ A’ B
A B’ A BA+B
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Three-variable map
Three-variable map has eight squares. Notice thatthe last two columns are not in numeric order. That isthe key idea that makes the map work.
0 A’B’C’ 2A’BC’
6ABC’
4AB’C’
1A’B’C
3A’BC
7ABC
5AB’C
CAB
0
1
00 01 11 10
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Three-variable map
Example: F(A,B,C)= Σ m(2,3,4,5)
0’ ’ ’
2’ ’
6’
4’ ’
CAB
0
00 01 11 10
1 1
1A’B’C
3A’BC
7ABC
5AB’C1 1 1
What is the simplified expression?
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Exercise
Plot the following on a Karnaugh map
F(a,b,c)= Σ m(0,1,3,6)Determine the simplified Boolean expressionin terms of sum-of- roducts usin K-ma .Show also algebraic manipulation forsimplification to verify your answer.
F(A,B,C)= Σ m(0,1,4,5)g= p’q’r+p’qr’+p’qr+pqr’+pq’r’
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Next meeting
Four-variable map
Product-of-sums simplification