CHAPTER 2 DBM20083 BOOLEAN ALGEBRA

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2.1 Boolean Function

Exercise 2 :Write the Boolean expression for out
put x for each of the following circuit diagrams.

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2.3 Minimization Of The Circuits

2.3.1 Define the minimization of the Circuits

 Once the expression for a logic circuit has been obtained,
we may be able to reduce it to a simpler form containing
fewer variables in one or more terms.

 The new expression can then be used to implement a circuit
that is equivalent to the original circuit but that contains
fewer gates and connections.

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2.3 Minimization Of The Circuits

2.3.2 Minimization of the Circuits – Karnaugh Map

 Karnaugh maps provide an alternative way of simplifying
logic circuits. Instead of using Boolean Algebra
simplification techniques, you can transfer logic values from
a Boolean statement or a truth table into a Karnaugh map.

 The arrangement of 0’s and 1’s within the map helps you to
visualize the logic relationships between the variables and
leads directly to a simplified Boolean statement.

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2.3 Minimization Of The Circuits

Karnaugh Map Format ത
ҧ 1 0
ABX 0 1
001
010 0 1
100 0
111 1
01
= ҧ ത +
10

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2.3 Minimization Of The Circuits

Karnaugh Map Format = ҧ ത ҧ + ҧ ത + ҧ ҧ + ҧ

ABCX ҧ 0 1
0001 ҧ ത 1 1 1 1
0011 ҧ 1 0 1 0
0101 00
0110
1000 01
1010
1101 1 0 11 1 0
1110 ത 0 0 10 0 0

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2.3 Minimization Of The Circuits

Looping

 The truth table has it 1’s and 0’s written into the Karnaugh
map in the corresponding locations, then each location in the
Karnaugh map represents one minterm. Then two, four or
eight adjacent squares that contain 1’s are looped to create
a simplified Boolean expression (note the number of
adjacent squares in a power of 2).

 The expression for the loop will eliminate the variable(s) that
appear in both normal and complemented form.

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2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

The Karnaugh map
uses the following
rules for the
simplification of
expressions by
grouping together
adjacent cells
containing ones.

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2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Karnaugh map –Rules of Simplification

 No zeros allowed
 No diagonals
 Only power of 2 number of cells in each group
 Groups should be as large as possible
 Every one must be in at least one group
 Overlapping allowed
 Wrap around allowed
 Fewest number of groups possible.

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following

expression.

) = ത +

ത Using algebraic simplification,
ҧ 0 0 = ത +
1 1 = ത +

=

=

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following

expression.

) = + ҧ + ҧ ത ) = + ҧ + ҧ ത + ത

ത ത
ҧ 1 1 ҧ 1 1
0 1 1 1

= ҧ + = 1

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following

expression.

) = ҧ ҧ + ҧ ) = ҧ ത ҧ + ത ҧ

ҧ ҧ
ҧ ത 0 0 ҧ ത 1 0
ҧ 1 0 ҧ 0 0


1 0 0 0

0 0 1 0
ത = ҧ ത = ത ҧ

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

) = ҧ ത + ҧ = + ത ) = ҧ ത ҧ + ҧ ത + ҧ + ҧ ҧ

ҧ ҧ
ҧ ത 0 1
ҧ 0 1 ҧ ത 1 1
ҧ 1 1

0 1
0 0
0 1
ത = 0 0
ത = ҧ

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

ℎ) = ҧ ത + ҧ ത + ҧ + ҧ ҧ + + ҧ

ҧ

ҧ ത 1 1

ҧ 1 1


1 1

0 ҧ 0
ത = +

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2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

) = ҧ ҧ + ത ҧ + ത + ҧ +

ҧ
ҧ ത
ҧ 1

1 1
1 1
ത = + ҧ

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

) = ҧ + ത + + ҧ +

ҧ
ҧ ത
ҧ 1

1 1
1
ത = + +

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

) = ҧ ത ҧ + ҧ + ҧ +

ҧ
ҧ ത 1
ҧ 1 1

1 1
1
ത = ҧ ҧ + +

Azlina Binti Morshidi/JMSK/PKS

2.3 Minimization Of The Circuits

Example : Use a K-map to simplify the following
expression.

) = ҧ ത ҧ + ҧ ത + ҧ ҧ + ҧ + ҧ + + ത ҧ + ത

ҧ
ҧ ത 1 1
ҧ 1 1

1 1
1 1
ത = 1

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) = ҧ ത + ത ) = + ҧ + ത
ത ത

ҧ 1 0 ҧ 0 1
1 0 1 1

= ത = +

) = ҧ ҧ + ҧ

ҧ

ҧ ത 0 0

ҧ 1 1


0 0

0 = ҧ 0
ത
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) = ҧ + ҧ ҧ + + ҧ ) = ҧ ത ҧ + ത ҧ + ҧ ҧ + ҧ

ҧ ҧ
ҧ ത 0 0 0
ҧ 1 1 ҧ ത 1 0
ҧ 1

1 1
1 0
0 0
ത = 1 0
ത = ҧ

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) = ҧ ത + ത ) = ҧ ത ҧ + ҧ ത + ത ҧ + ത

ҧ ҧ
ҧ ത 0 1 1
ҧ 0 0 ҧ ത 1 0
ҧ 0

0 0
0 0
0 1
ത = ത 1 1
ത = ത

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ℎ) = ҧ + ҧ + + ത ҧ )
= ҧ ത ҧ + ҧ ത + ҧ + ҧ ҧ
ҧ + ത ҧ + ҧ
ҧ ത 0 0
ҧ 1 1 ҧ

ҧ ത 1 1
1 1 ҧ 1 1

1 0
ത = + ҧ 1 0

1 0
ത = ҧ + ҧ

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Exercise

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Exercise

Azlina Binti Morshidi/JMSK/PKS

Exercise

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ҧ

Azlina Binti Morshidi/JMSK/PKS

Azlina Binti Morshidi/JMSK/PKS

Azlina Binti Morshidi/JMSK/PKS

) = ҧ ത + ത ) = ҧ ҧ ҧ + ҧ
ത

ҧ ҧ ത

ҧ
= ത




ത = ҧ

Azlina Binti Morshidi/JMSK/PKS

) = ҧ ത + ത ) = ҧ ҧ ҧ + ҧ
ത

ҧ ҧ ത

ҧ
= ത




ത = ҧ

Azlina Binti Morshidi/JMSK/PKS

) = ҧ ത + ത ) = ҧ ҧ ҧ + ҧ
ത

ҧ ҧ ത

ҧ
= ത




ത = ҧ

Azlina Binti Morshidi/JMSK/PKS

) = ҧ ത + ത ) = ҧ ҧ ҧ + ҧ
ത

ҧ ҧ ത

ҧ
= ത




ത = ҧ

Azlina Binti Morshidi/JMSK/PKS

) = ҧ ത + ത ) = ҧ ҧ ҧ + ҧ
ത

ҧ ҧ ത

ҧ
= ത




ത = ҧ

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Azlina Binti Morshidi/JMSK/PKS