1
KARNAUGH MAP
•
•
•
•
•
Introduction
Strategy for Minimization
Minimization of Product-of-Sums Forms
Minimization of More Complex Expressions
Don't care Terms
2
Introduction
• Why karnaugh map
• Example (With Boolean algebra)
W=A+ .B
=A.(B+ )+ .B
=A.B+A. + .B
= A . ( B + ) + B ( A+
=A+B
)
3
Introduction ( cont. )
• Using Boolean algebra for minimization causes it’s own
problem because of it mainly being a trial and error process,
and we can almost never be sure that we have reached a
minimal representation.
• If we can form a graphical notation for our Boolean algebra
the insight need for the minimization will be less vital in
solving the problems.
We can come close to our aim by using a graphical
notation named Karnaugh Map that will be defined in
next slides
4
As it can be seen,
each box of the
Karnaugh map
corresponds to a row
of the truth table and
been numbered
Booleanhas
Algebra
accordingly
Introduction ( cont. )
• Comparing Karnaugh Map and
Truth Table
Karnaugh Map
A B
W
0
0
0
0
1
1
1
0
1
1
1
1
W = A. B + A . B + A . B =
W = A . B + A . B + A . B + A .B=
W= B ( A + A ) + A (B + B ) = A + B
A 0
B
0
1
0
1
1
1
1
W
This form of
representing w in
the following
example is called a
Sum of Product
(SOP)
Which will be
define in next slides
5
Strategy for Minimization
• Terminology
• Minimization Procedure
6
Terminology
• Implicant : Product term that implies function
• Prime Implicant : An Implicant that is not completely
covered by any other Implicant but itself
• Essential prime Implicant : A prime Implicant that has a
minter not covered by any other prime Implicant
• Product term : An and expression
7
Terminology
• Minterm : We define a Minterm to be a product that contains all
variables of that particular switching function in either
complemented or non-complemented form
• Maxterm : We define a Maxterm to be a sum that contains all
variables of that particular switching function in either
complemented or non-complemented form
• Standard SOP(Sum Of Products) : In standard SOP, the products
are obtained directly from the Karnaugh map or truth table, so
the SOP contains all of the variables of the function
• Standard POS(Product Of Sums) : In standard POS, the products
are obtained directly from the Karnaugh map or truth table, so
the POS contains all of the variables of the function
8
Terminology ( cont. )
• A simpler shorthand form of representing a SOP is to use
the number of the Minterms that appear in that
representation. In the following example for instance we
could have written
Karnaugh Map
C
AB
0
0
1
4
0
0
1
1
5
W=
01
0
1
3
7
11
0
0
2
10
1
6
𝑚(2,4,5,6)
1
9
Terminology ( cont. )
• Sometimes writing an expression in a POS form is easier as
seen in the following example:
Karnaugh Map
C
AB
00
1
W=
𝑚(1,3,4,5,6,7)
4
00
0
1
1
5
01
1
1
3
7
11
1
1
w=
2
6
10
0
1
𝑀 0,2 = (a + b + c) . (𝑎 + b + c)
10
Strategy for Minimization
• Terminology
• Minimization Procedure