6_boolean expression simplification

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Boolean ExpressionSimplification

Algebra Method

Map Method



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.



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.



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.



Types of Maps- According to number of variables

Two-Variable MapThree-Variable Map

-



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



Two-variable mapExample : f(A,B)= Σ m(0,3)

1B

A

0

0 1What is the algebraic

11 the map?



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



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



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



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?



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’



Next meeting

Four-variable map

Product-of-sums simplification

6_boolean expression simplification

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