Chapter 3_Logic circuits

CE211 Digital SystemsCE211 Digital SystemsGate Level Minimization

11 / / 143143

KarnaughKarnaugh

MapMap

Adjacent Squares

Number of squares = number of combinations

Each square represents a minterm

2 Variables 4 squares



Each two adjacent squares differ in one variable

Two adjacent minterms

can be combined together

Example: F = x y + x y

= x ( y + y )

= x

22 / / 143143

TwoTwo--variable Mapvariable Map

m0 m1

m2 m3

x y Minterm

0 0 0 m01 0 1 m12 1 0 m23 1 1 m3

yxyxyxyx

yx 0 1

0

1

yx yx

yx yx

Note: adjacent squares horizontally and vertically NOT diagonally

33 / / 143143


Examplex y F Minterm

0 0 0 0 m01 0 1 0 m12 1 0 0 m23 1 1 1 m3

yx 0 1

0

1

m0 m1

m2 m3

y

0 0

x 0 1

yxyxyxyx

yx yx

yx yx

44 / / 143143


Examplex y F Minterm

0 0 0 0 m01 0 1 1 m12 1 0 1 m23 1 1 1 m3

yx 0 1

0

1

m0 m1

m2 m3

y

0 1

x 1 1

yxyxyxF ++=

yxx )( + )( yyx +xyF +=

yxyxyxyx

yx yx

yx yx

55 / / 143143

ThreeThree--variable Mapvariable Map

m0 m1 m3 m2

m4 m5 m7 m6

x y z Minterm

0 0 0 0 m01 0 0 1 m12 0 1 0 m23 0 1 1 m34 1 0 0 m45 1 0 1 m56 1 1 0 m67 1 1 1 m7

zyx

y zx 00 01 11 10

0

1

zyxzyxzyxzyxzyxzyxzyx

zyx zyx zyxzyx

zyx zyx zyxzyx

66 / / 143143


m0 m1 m3 m2m4 m5 m7 m6x y z F Minterm

0 0 0 0 0 m01 0 0 1 0 m12 0 1 0 1 m23 0 1 1 1 m34 1 0 0 1 m45 1 0 1 1 m56 1 1 0 0 m67 1 1 1 0 m7

y zx 00 01 11 10

0

1

Example

y

0 0 1 1

x 1 1 0 0z

=F yxyx +

zyxzyxzyxzyxzyxzyxzyxzyx

zyx zyx zyxzyxzyx zyx zyxzyx

77 / / 143143yx



0 0 0 0 0 m01 0 0 1 0 m12 0 1 0 0 m23 0 1 1 1 m34 1 0 0 1 m45 1 0 1 0 m56 1 1 0 1 m67 1 1 1 1 m7

y zx 00 01 11 10

0

1

Example

y

0 0 1 0

x 1 0 1 1z

=F zyzx + +Extra



88 / / 143143


x y z F Minterm

0 0 0 0 0 m01 0 0 1 1 m12 0 1 0 0 m23 0 1 1 1 m34 1 0 0 0 m45 1 0 1 1 m56 1 1 0 0 m67 1 1 1 1 m7

Example y0 1 1 0

x 0 1 1 0z

zx zx

zyxzyxzyxzyxF +++=

)( yyzx + )( yyzx +

y

0 1 1 0

x 0 1 1 0z


z

99 / / 143143



0 0 0 0 1 m01 0 0 1 0 m12 0 1 0 1 m23 0 1 1 0 m34 1 0 0 1 m45 1 0 1 1 m56 1 1 0 1 m67 1 1 1 0 m7

y zx 00 01 11 10

0

1

Example

y

1 0 0 1

x 1 1 0 1z

=F z yx+



1010 / / 143143

FourFour--variable Mapvariable Map

m0 m1 m3 m2m4 m5 m7 m6m12 m13 m15 m14m8 m9 m11 m10

w x y z Minterm0 0 0 0 0 m01 0 0 0 1 m12 0 0 1 0 m23 0 0 1 1 m34 0 1 0 0 m45 0 1 0 1 m56 0 1 1 0 m67 0 1 1 1 m78 1 0 0 0 m89 1 0 0 1 m9

10 1 0 1 0 m1011 1 0 1 1 m1112 1 1 0 0 m1213 1 1 0 1 m1314 1 1 1 0 m1415 1 1 1 1 m15

y zwx 00 01 11 10

00

01

11

10

zyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxw

zyxw zyxw yzxw zyxw

zyxw zyxw xyzw zxyw

zyxw zyxw yzxw zyxw

zywx zywx wxyz zwxy

1111 / / 143143


w x y z F Minterm0 0 0 0 0 1 m01 0 0 0 1 1 m12 0 0 1 0 1 m23 0 0 1 1 0 m34 0 1 0 0 1 m45 0 1 0 1 1 m56 0 1 1 0 1 m67 0 1 1 1 0 m78 1 0 0 0 1 m89 1 0 0 1 1 m910 1 0 1 0 0 m1011 1 0 1 1 0 m1112 1 1 0 0 1 m1213 1 1 0 1 1 m1314 1 1 1 0 1 m1415 1 1 1 1 0 m15

y zwx 00 01 11 10

00011110

zyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxwzyxw

zyxw zyxw yzxw zyxwzyxw zyxw xyzw zxyw

zyxw zyxw yzxw zyxwzywx zywx wxyz zwxy

Example

y1 1 0 11 1 0 1

xw

1 1 0 11 1 0 0

z

=F y zw+ + zx

1212 / / 143143


ExampleSimplify: F = A B C + B C D + A B C D + A B C

C

BA

D

1313 / / 143143



C

BA

D

1 1

1414 / / 143143



C

BA

D

1

1

1515 / / 143143



C

BA

D

1

1616 / / 143143



C

BA

D1 1

1717 / / 143143



C1 1 1

1B

A1 1 1

D

=F DB CB+ + DCA

1818 / / 143143

FiveFive--variable Mapvariable Map

DE DBC 00 01 11 10

00 m0 m1 m3 m2

01 m4 m5 m7 m6C

B11 m12 m13 m15 m14

10 m8 m9 m11 m10

E

A = 0

DE DBC 00 01 11 10

00 m16 m17 m19 m18

01 m20 m21 m23 m22C

B11 m28 m29 m31 m30

10 m24 m25 m27 m26

E

A = 1

1919 / / 143143

FiveFive--variable Mapvariable Map

A = 0

A = 1

2020 / / 143143

ImplicantsImplicants

C

1

1 1 1B

A1 1 1

1

D

Implicant:Gives F = 1

2121 / / 143143

Prime Prime ImplicantsImplicants

C

1

1 1 1B

A1 1 1

1

D

Prime Implicant:Cant grow beyond this size

2222 / / 143143

Essential Prime Essential Prime ImplicantsImplicants

C

1

1 1 1B

A1 1 1

1

D

Essential Prime Implicant:No other choice

Not essential

8 Implicants5 Prime

implicants

4 Essential

prime

implicants

2323 / / 143143

Product of Sums SimplificationProduct of Sums Simplification

w x y z F F0 0 0 0 0 1 01 0 0 0 1 1 02 0 0 1 0 1 03 0 0 1 1 0 14 0 1 0 0 1 05 0 1 0 1 1 06 0 1 1 0 1 07 0 1 1 1 0 18 1 0 0 0 1 09 1 0 0 1 1 010 1 0 1 0 0 111 1 0 1 1 0 112 1 1 0 0 1 013 1 1 0 1 1 014 1 1 1 0 1 015 1 1 1 1 0 1

y1 1 0 11 1 0 1

xw

1 1 0 11 1 0 0

z =Fy zw+ + zx

y1 1 0 11 1 0 1

xw

1 1 0 11 1 0 0

z

=F zy + yxw

yxwzyF +=

)()( yxwzyF +++=

F

ywz

zx

y

wzxy

2424 / / 143143

DonDontt--Care ConditionCare Condition

Example

otherwiseC

0depositedisnicleaif1{=

otherwiseB

0depositedisdimeaif1{=

otherwiseA

0depositedisquarteraif1{=

A B C $ Value0 0 0 $ 0.000 0 1 $ 0.050 1 0 $ 0.100 1 1 Not possible1 0 0 $ 0.251 0 1 Not possible1 1 0 Not possible1 1 1 Not possible

You can only drop one coin at

a time.

Used as dont care
http://www.vending101.com/snacks.htm

2525 / / 143143


Example

A B C F0 0 0 00 0 1 10 1 0 10 1 1 x1 0 0 11 0 1 x1 1 0 x1 1 1 x

F

Dont care what value F may take

Logic Circuit

=

=

)7,6,5,3(),,(

)4,2,1(),,(

CBAd

CBAF

A

B

C
http://www.vending101.com/snacks.htm

2626 / / 143143


Example

F

B

0 1 x 1

A 1 x x xC

BCAFCBACBACBAF

++=++=

A

B

C

2727 / / 143143


Example

y

x 1 1 x

x 1x

w1

1z

F (w, x, y, z) = (1, 3, 7, 11, 15)

d (w, x, y, z) = (0, 2, 5)

zwzyF +=

x = 0

x = 1

y

x x

0 x 0x

w0 0 0

0 0 0z

ywzF +=

x = 0x = 1

2828 / / 143143

QuineQuineMcCluskey Tabular Method MotivationMcCluskey Tabular Method Motivation

Karnaugh maps are effective for the minimizationof expressions with up to 5 or 6 inputs

difficult to use and error prone for circuits with many inputs.

Karnaugh maps depend on our ability to visually identify prime implicants and select a set of prime implicants that cover all minterms.

They do not provide a direct algorithm to be implemented in a computer.

For larger systems, we need a programmable method!!

2929 / / 143143

QuineQuine--McCluskeyMcCluskey

Quine, Willard, A way to simplify truth functions.American Mathematical Monthly, vol. 62, 1955.

Quine, Willard, The problem of simplifying truth functions.American Mathematical Monthly, vol. 59, 1952.

Willard van Orman Quine 1908-2000, Edgar Pierce Chair of Philosophy at Harvard University.http://members.aol.com/drquine/wv-quine.html

McCluskey Jr., Edward J. Minimization of Boolean Functions.Bell Systems Technical Journal, vol. 35, pp. 1417-1444, 1956

Edward J. McCluskey, Professor of ElectricalEngineering and Computer Science at Stanfordhttp://www-crc.stanford.edu/users/ejm/McCluskey_Edward.html

3030 / / 143143

Outline of the QuineOutline of the Quine--McCluskey MethodMcCluskey Method

1. Produce a minterm expansion (standard sum-of-products form) for a function F

2. Eliminate as many literals as possible bysystematically applying XY + XY

= X.

3. Use a prime implicant chart to select aminimum set of prime implicants

that

when ORed together produce F, and thatcontains a minimum number of literals.

3131 / / 143143

Determination of Prime ImplicantsDetermination of Prime Implicants

ABCD

+ ABCD = ABC

1 0 1 0 + 1 0 1 1 = 1 0 1 -(The dash indicates a missing variable)

ABCD + ABCD

0 1 0 1 + 0 1 1 0

We can combine the minterms above because theydiffer by a single bit.

The minterms below wont combine

3232 / / 143143

QuineQuine--McCluskey Method An ExampleMcCluskey Method An Example

1. Find all the prime implicants

= )14,10,9,8,7,6,5,2,1,0(),,,( mdcbaf

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Group the minterms according to the numberof 1s in the minterm.

This way we only have tocompare minterms fromadjacent groups.

3333 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

Combininggroup 0 and

group 1:

3434 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-


group 1:

3535 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-0


group 1:

3636 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -000

Does it makesense tocombine group 0with group 2 or 3?

No, there are atleast two bits thatare different.

3737 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -000

Does it makesense to nocombine group 0with group 2 or 3?

No, there are atleast two bits thatare different.

Thus, next we combine group 1and group 2.

3838 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-01

Combine group 1and group 2.

3939 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-01


4040 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1

and group 2.

4141 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1

and group 2.

4242 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1

and group 2.

4343 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -001Combine group 1

and group 2.

4444 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10


4545 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10


4646 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -010


4747 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -010


4848 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -010


4949 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-


5050 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-0

Again, there isno need to tryto combine group1 with group 3.

Lets try to combinegroup 2 with group 3.

5151 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-1


5252 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-1


5353 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-


5454 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -110

Combine

group 2and group 3.

5555 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -110


5656 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -110


5757 / / 143143


group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -110


5858 / / 143143


Column I Column II

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

We have nowcompleted thefirst step. Allminterms in column I wereincluded.

We can dividecolumn II intogroups.

5959 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

6060 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

6161 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

6262 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

6363 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

6464 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

6565 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

6666 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

6767 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

6868 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

6969 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

7070 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

7171 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-

7272 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

7373 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

7474 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-

7575 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-

7676 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

7777 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

7878 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

7979 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

8080 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

8181 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --10

8282 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --102,10,6,14 --10

8383 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --102,10,6,14 --10

8484 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --102,10,6,14 --10

8585 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --102,10,6,14 --10

No more combinationsare possible, thus westop here.

8686 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

0,8,1,9 -00-0,8,2,10 -0-0

2,6,10,14 --102,10,6,14 --10

We can eliminate repeatedcombinations

8787 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd

Now we form f with theterms not checked

8888 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd + abd

8989 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd + abd + abc

9090 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd + abd + abc + bc

9191 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd + abd + abc + bc+ bd

9292 / / 143143


Column I Column II

group 0

group 1

group 2

group 3

0 0000

1 00012 00108 1000

5 01016 01109 100110 1010

7 011114 1110

0,1 000-0,2 00-00,8 -0001,5 0-011,9 -0012,6 0-10

2,10 -0108,9 100-

8,10 10-05,7 01-16,7 011-

6,14 -11010,14 1-10

Column III

0,1,8,9 -00-0,2,8,10 -0-0

2,6,10,14 --10

f = acd + abd + abc + bc+ bd

+ cd

9393 / / 143143



+ bd

+ cd

But, the form below is not minimized, using a Karnaugh map we can obtain:

a

1

b

c

d1

9494 / / 143143



+ bd

+ cd


a

1

1

b

c

d1

9595 / / 143143



+ bd

+ cd


a

1

1

1

b

c

d1

9696 / / 143143



+ bd

+ cd


a

1

1

1

1

1

1

b

c

d1

9797 / / 143143



+ bd

+ cd


a

1

1

1

1

1

1

1 1

b

c

d1

9898 / / 143143



+ bd

+ cd

But, the form below is not minimized. Using a Karnaugh map we can obtain:

a

1

1

1

1

1

1

1 1 1

b

c

d1

9999 / / 143143


+ bd

+ cd


a

1

1

1

1

1

1

1 1 1

b

c

d1

F = abd


100100 / / 143143


+ bd

+ cd


a

1

1

1

1

1

1

1 1 1

b

c

d1

F = abd + cd


101101 / / 143143


+ bd

+ cd


a

1

1

1

1

1

1

1 1 1

b

c

d1

F = abd + cd

+ bc


102102 / / 143143



+ bd

+ cd

What are the extra terms in the solution obtainedwith the Quine-McCluskey method?

a

1

1

1

1

1

1

1 1 1

b

c

d1

F = abd + cd

+ bc

Thus, we need a method to eliminate this redundant termsfrom the Quine-McCluskey solution.

103103 / / 143143

The Prime Implicant ChartThe Prime Implicant Chart

The prime implicant chart is the second part ofthe Quine-McCluskey procedure.

It is used to select a minimum set of prime implicants.

Similar to the Karnaugh map, we first selectthe essential prime implicants, and then weselect enough prime implicants to cover allthe minterms of the function.

104104 / / 143143

Prime Implicant Chart (Example)Prime Implicant Chart (Example)

0 1 2 5 6 7 8 9 10 14(0,1,8,9) bc X X X X (0,2,8,10) bd X X X X (2,6,10,14) cd X X X X (1,5) acd X X (5,7) abd X X (6,7) abc X X

Question: Given the prime implicant chart above,how can we identify the essential primeimplicants of the function?

mintermsP

rime

Impl

ican

ts

105105 / / 143143



Similar to the Karnaugh map, all we have to do is to look for minterms that are covered by a singleterm.

mintermsP

rime

Impl

ican

ts

106106 / / 143143



Once a term is included in the solution, all theminterms covered by that term are covered.

Therefore we may now mark the covered mintermsand find terms that are no longer useful.

mintermsP

rime

Impl

ican

ts

107107 / / 143143



mintermsP

rime

Impl

ican

ts

108108 / / 143143



As we have not covered all the minterms withessential prime implicants, we must chooseenough non-essential prime implicants to cover the remaining minterms.

mintermsP

rime

Impl

ican

ts

109109 / / 143143



What strategy should we use to find a minimumcover for the remaining minterms?

mintermsP

rime

Impl

ican

ts

110110 / / 143143



We choose first prime implicants that cover themost minterms. Should this strategy always work??

mintermsP

rime

Impl

ican

ts

111111 / / 143143



Therefore our minimum solution is:

f(a,b,c,d) = bc

+ cd

+ abd

mintermsP

rime

Impl

ican

ts

112112 / / 143143

Cyclic Prime Implicant ChartCyclic Prime Implicant Chart

F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-

0 1 2 5 6 7(0,1) ab X X (0,2) ac X X (1,5) bc X X (2,6) bc X X (5,7) ac X X(6,7) ab X X

Which ones are the essential prime implicants in this chart?

There is no essential prime implicants, how we proceed?

minterms

Prim

e Im

plic

ants

113113 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-


Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.

minterms

Prim

e Im

plic

ants

F(a,b,c) = ab

114114 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-


Also, all implicants cover the same number of minterms. We will have to proceed by trial and error.

minterms

Prim

e Im

plic

ants

F(a,b,c) = ab

+ bc

115115 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-


Thus, we get the minimization:

F(a,b,c) = ab

+ bc

+ ac

minterms

Prim

e Im

plic

ants

116116 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-


Lets try another set of prime implicants.

minterms

Prim

e Im

plic

ants

117117 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-



minterms

Prim

e Im

plic

ants

F(a,b,c) = ac

118118 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-



minterms

Prim

e Im

plic

ants

F(a,b,c) = ac + bc

119119 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-



minterms

Prim

e Im

plic

ants

F(a,b,c) = ac + bc+ ab

120120 / / 143143


F(a,b,c) =

m(0, 1, 2, 5, 6, 7)

0 000 1 001 2 010 5 101 6 110 7 111

0,1 00-0,2 0-01,5 -012,6 -105,7 1-16,7 11-


This time we obtain:


minterms

Prim

e Im

plic

ants

121121 / / 143143


Which minimal form is better?

F(a,b,c) = ab

+ bc

+ ac


Depends on what terms we must form for otherfunctions that we must also implement.

Often we are interested in examining all minimalforms for a given function.

Thus we need an algorithm to do so.

122122 / / 143143

PetrickPetricks Methods Method

S. R. Petrick. A direct determination of the irredundantforms of a boolean function from the set of prime implicants. Technical Report AFCRC-TR-56-110, Air Force Cambridge Research Center, Cambridge, MA, April, 1956.

Goal: Given a prime implicant chart, determineall minimum sum-of-products solutions.

123123 / / 143143

PetrickPetricks Methods Method An ExampleAn Example

0 1 2 5 6 7P1 (0,1) ab X X P2 (0,2) ac X X P3 (1,5) bc X X P4 (2,6) bc X X P5 (5,7) ac X XP6 (6,7) ab X X

Step 1: Label all the rows in the chart.

Step 2: Form a logic function P with the logic variables P1

, P2

, P3

that is true whenall the minterms in the chart are covered.

minterms

Prim

e Im

plic

ants

124124 / / 143143



The first column has an X in rows P1

and P2

. Therefore we must include one of these rowsin order to cover minterm 0. Thus the followingterm must be in P:

(P1

+ P2

)

minterms

Prim

e Im

plic

ants

125125 / / 143143



Following this technique, we obtain:

P = (P1

+ P2

) (P1

+ P3

) (P2

+ P4

) (P3

+ P5

) (P4

+ P6

) (P5

+ P6

)

P = (P1

+ P2

) (P1

+ P3

) (P4

+ P2

) (P5

+ P3

) (P4

+ P6

) (P5

+ P6

)

P = (P1

+ P2

) (P1

+ P3

) (P4

+ P2

) (P4

+ P6

) (P5

+ P3

) (P5

+ P6

)

P = (P1

+ P2 P3

) (P4

+ P2 P6

) (P5

+ P3 P6

)

minterms

Prim

e Im

plic

ants

126126 / / 143143


P = (P1

+ P2

) (P1

+ P3

) (P2

+ P4

) (P3

+ P5

) (P4

+ P6

) (P5

+ P6

)

P = (P1

+ P2

) (P1

+ P3

) (P4

+ P2

) (P5

+ P3

) (P4

+ P6

) (P5

+ P6

)

P = (P1

+ P2

) (P1

+ P3

) (P4

+ P2

) (P4

+ P6

) (P5

+ P3

) (P5

+ P6

)

P = (P1

+ P2 P3

) (P4

+ P2 P6

) (P5

+ P3 P6

)

P = (P1 P4

+ P1

P2 P6

+ P2 P3 P4

+ P2 P3 P6

) (P5

+ P3 P6

)

P = P1 P4 P5

+ P1

P2 P5 P6

+ P2 P3 P4 P5

+ P2 P3 P5 P6+ P1 P3 P4 P6

+ P1

P2 P3 P6

+ P2 P3 P4 P6

+ P2 P3 P6

P = P1 P4 P5

+ P1

P2 P5 P6

+ P2 P3 P4 P5

+ P1 P4 P3 P6

+ P2 P3 P6

127127 / / 143143

PetrickPetricks Methods Method

An ExampleAn Example

P = P1 P4 P5

+ P1

P2 P5 P6

+ P2 P3 P4 P5

+ P1 P4 P3 P6

+ P2 P3 P6

This expression says that to cover all the mintermswe must include the terms in line P1

and line P4

and line P5

, or we must include line P1

, and line P2

, and line P5

, and line P6

, or

Considering that all the terms P1

, P2

,

have the samecost, how many minimal forms the function has?

The two minimal forms are P1 P4 P5 and P2 P3 P6:

F = ab

+ bc

+ ac F = ac

+ bc + ab

128128 / / 143143

Universal GatesUniversal Gates

One Type

Use as many as you need (quantity), but one type only.

Perform Basic Operations

AND, OR, and NOT

NAND Gate

NOT-AND functions

OR function can be obtained from AND by Demorgans

NOR Gate

NOT-OR functions (AND by Demorgans)

129129 / / 143143


NAND Gate

NOT:

AND:

OR: DeMorgans

130130 / / 143143


NOR Gate

NOT:

OR:

AND: DeMorgans

131131 / / 143143

NAND & NOR ImplementationNAND & NOR Implementation

Two-Level Implementation

132132 / / 143143


Two-Level Implementation

133133 / / 143143


Multilevel NAND Implementation

CDBABC

F

CDBABC

F

134134 / / 143143


Multilevel NOR Implementation

135135 / / 143143

Gate ShapesGate Shapes

AND

OR

NAND

NOR

136136 / / 143143

Other ImplementationsOther Implementations

AND-OR-Invert

OR-AND-Invert

137137 / / 143143

Implementations SummaryImplementations Summary

Sum Of Products:

AND-OR

AND-OR-Invert ==

AND-NOR ==

NAND-AND

Products Of Sums

OR-AND

OR-AND-Invert ==

OR-NAND ==

NOR--OR

138138 / / 143143

ExclusiveExclusive--OROR

XORF = x

y = x y + x y

XNORF = x

y =

x y = x y + x y

139139 / / 143143

ExclusiveExclusive--OROR

Identities

x

0 =

x

x

1 =

x

x

x =

0

x

x =

1

x

y =

x

y =

x

y

Commutative & Associative

x

y =

y

x

( x

y )

z =

x

( y

z ) =

x

y

z

x y XOR0 0 00 1 11 0 11 1 0

140140 / / 143143

ExclusiveExclusive--OR FunctionsOR Functions

Odd FunctionF = x

y

z

F = (1, 2, 4, 7)

Even FunctionF = x

y

z

F = (0, 3, 5, 6)

x y z XOR XNOR0 0 0 0 10 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 0

y zx 00 01 11 10

0 0 1 0 1

1 1 0 1 0

141141 / / 143143

ParityParity

1010

1010

1000

10101

10101

10001

Parity Generator

Parity Checker

142142 / / 143143

Parity GeneratorParity Generator

Odd Parity

Even Parity

1010

1

Odd number of 1s

1010

0

Even number of 1s

1010

1010

143143 / / 143143

Parity CheckerParity Checker

Odd Parity

Even Parity

ErrorCheck

1010

1

1010

0Error

Check
CE211 Digital SystemsKarnaugh MapTwo-variable MapTwo-variable MapTwo-variable MapThree-variable MapThree-variable MapThree-variable MapThree-variable MapThree-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFour-variable MapFive-variable MapFive-variable MapImplicantsPrime ImplicantsEssential Prime ImplicantsProduct of Sums SimplificationDont-Care ConditionDont-Care ConditionDont-Care ConditionDont-Care ConditionQuineMcCluskey Tabular Method MotivationQuine-McCluskeyOutline of the Quine-McCluskey MethodDetermination of Prime ImplicantsQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleQuine-McCluskey Method An ExampleSlide Number 100Slide Number 101Slide Number 102Quine-McCluskey Method An ExampleThe Prime Implicant ChartPrime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Prime Implicant Chart (Example)Cyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartCyclic Prime Implicant ChartPetricks MethodPetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExamplePetricks MethodAn ExampleUniversal GatesUniversal GatesUniversal GatesNAND & NOR ImplementationNAND & NOR ImplementationNAND & NOR ImplementationNAND & NOR ImplementationGate ShapesOther ImplementationsImplementations SummaryExclusive-ORExclusive-ORExclusive-OR FunctionsParityParity GeneratorParity Checker

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Chapter 3_Logic circuits