The document provides information on logic minimization techniques including Karnaugh maps (K-maps) for two, three, four and more variables, prime implicant charts, don't care conditions, NAND and NOR gate implementations, and the Quine-McCluskey (Q-M) tabulation method. Examples are given for each topic to demonstrate how to use the techniques to minimize logic functions and implement them using basic gates.

1. m0 m1
m3m2
(a)
y
x’y’ x’y
xyxy’1
0
y
x 0 1
x
(b)
(b) x+y
1
0
y
x 0 1
x
y
(a) xy
1
0
y
x 0 1
x
y
1
1
11
TWO VARIABLE K-MAP
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2. m0
m4
m1 m3 m2
m5 m6m7
y
x
x’y’z’
xy’z’
x’y’z x’yz x’yz’
xy’z xyz’xyz
yz
x
0
1
00 01 11 10
z
THREE VARIABLE MAP
y
z
yz
1
0
1 1
1 1
1 1
1 1
00 01 11 10 MAP FOR
F=x’yz+x’yz’+xy’z’+xy’z
F=x’y+xy’
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3. Reflected Code
Binary Code Reflected Code
0000 0000
0001 0001
0010 0011
0011 0010
0100 0110
0101 0111
0110 0101
0111 0100
1000 1100
1001 1101
1010 1111
1011 1110
1100 1010
1101 1011
1110 1001
1111 1000
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4. MAP FO R : x’yz+xy’z’+xyz+xyz’=yz+xz’
yz
x
00 01 11 10
0
1 1
1
1
1
z
y
A
BC
A
00 01 11 10
0
1
C
BC B
1
1 1
11
1
MAP FOR : A’C+A’B+AB’C+BC=C+A’B
yz
x
00 01 11 10
0
1
y
x
F(x,y,z)=Σ(0,2,4,5,6)=z’+xy’
1
1
1 1
1
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5. m0 m1 m3 m2
m6
m14
m10m11m9m8m8
m12 m13 m15
m7m5m4
(a)
y
z
w
00
01
11
10
00 01 11 10wx
yz
x
w’x’y’z’
w’xy’z’
wxy’z’
wx’y’z’
w’x’y’z w’x’yzw’x’yz w’x’yz’
w’xyz’
wxyz’
wx’yz’wx’yzwx’y’z
wxy’z wxyz
w’xyzw’xy’z
FOUR VARIBALE MAP
1 square represents a term of 4 literals.
2 adjacent squares represent a term of 3 literals.
4 adjacent squares represent a term of 2 literals.
8 adjacent squares represent a term of 1 literal.
16 adjacent squares represent the function equal to 1
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6. MAP FOR : F(w,x,y,z)=Σ(0,1,2,4,5,6,8,9,12,13,14)=y’+w’z’+xz’
y
yz
z
00
01
11
10
10110100
wx
x
w
1
1
1
1 1
1
1
1
1
1
1
C
CD
D
00
01
11
10
10110100
AB
B
A
1 1 1
1
1
MAP FOR : A’B’C’+B’CD’+A’BCD’+AB’C’=B’D’+B’C’+A’CD’
1 1
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7. A
C
AB
00
01
11
10
000 001 011 010 110 111 101 100
CDE
B
D
E
E
FIVE -VARIABLE MAP
0 1 3 2 6 77 5 4
8 9 11 10 15 13 12
282931
==
30
14
26272524
16 17 19 18 22 23 21 20
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8. B
ABC
DEF
000 001 011 010 110 111 101 100
000
001
011
010
0 1 3 2 6 77 5 4
8 9 11 10 15 13 12
282930
31
26272524
16 17 19 18 22 23 21 20
110
111
111
101
100
C
C
D
A
E FF
48
56
49 51 50 54 55 53 52
57 59 58 62 63 61 60
40 41 43 42 46 47 45 44
32 33 35 34 38 39 37 36
SIX VARIABLE MAP
14
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9. No of adjacent squares & literals in a term
2k
adjacent squares=n-k literals
n-variable map
k 2k
n=2 n=3 n=4 n=5 n=6 n=7
0 1 2 3 4 5 6 7
1 2 1 2 3 4 5 6
0 4 0 1 2 3 4 5
3 8 0 1 2 3 4
4 16 0 1 2 3
5 32 0 1 2
6 64 0 1
7 128 0FaaDoOEngineers.com

10. 11
1111
1111
11
1 1
CDE
000 001 011 010 110 111 101 100
C
B
AB
00
01
11
10
E D E
A
F=BE+AD’E+A’B’E’
F(A,B,C,D,E)=Σ
(0,2,4,6,9,11,13,15,17,21,25,27,29,31)
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11. Example
F(A , B, C, D) =∑(0,1,2,5,8,9,10)
(ii) Simplify it in:
(iii) Sum of products
(iv) Product of sums
AB
(i) Sum of products
F=B’D’+B’C’+A’C’D
F’ =AB+CD+BD’
(ii) F=(A’+B’)(C’+D’)(B’+D)
00
01
11
10
A
11
CD
00 01 10
1011
0000
0010
1011
C
D
B
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12. B’
D’
C’
A’
D
F
A’
B’
D’
C’
D
F
F=B’D’+B’C’+A’C’D F=(A’+B’)(C’+D’)(B’+D)
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13. GATE IMPLEMENTATION
X Y Z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 1
7 0 1 0
1 1 0 1
1 1 1 0
TRUTH TABLE OF
FUNCTION F
F(x,y,z)= Σ(1,3,4,6)
F(x,y,z)= Σ(1,3,4,6)
F(x,y,z) = ∏ (0,2,5,7)
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14. 1001
0110
0
1
00 01 11 10
z
yzx
x
y
Sum of products
F=x’z+xz’)
Product of sums
F’=xz+x’z’
F=(x’+z’)(x+z)
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15. NAND and NOR implementation
x
y
z
F=(xyz)’
AND invert
x
y
z
F=(x’+y’+z’)=(xyz)’
Invert-OR
F=(x+y+z)’
OR-invert
(a) TWO GRAPHIC SYMBOLS FOR NAND GATE
x
y
z F=x’y’z’=(x+y+z)’
Invert-AND
(b) TWO GRAPHIC SYMBOLS FOR NOR GATE
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16. x’x x x’
Buffer-invert AND-invert
x x’
OR-invert
THREE GRAPHIC SYMBOLS FOR INVERTORFaaDoOEngineers.com

17. A
B
C
D
E
A
B
C
D
E’
F
F
(a)
(b)
NAND Implementation
Simplified Boolean
function
F=AB+CD+E
Convert AND gates
with at least two
literals to NAND
gates at first level.
THREE WAYS TO IMPLEMENT
F=AB+CD+E
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18. Draw a single NAND
gate at level-2
&
NAND gate for a
term with single
literal
(c)
A
B
C
D
E
F
F={(AB)’.(CD)’.E’}
F=AB+CD+D
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19. x’
y’
z’
x
F
Two-level implementation
1000
0001
z
00 01 11 10
yyz
0
1
x F(x,y,z)= Σ(0,6)
F=x’y’z’+xyz’
F’=x’y+xy’+z
(a) MAP SIMPLIFICATION IN SUM OF PRODUCTS
y
z’
(b) F=x’y’z’+xyz’
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20. X’
y
x
Y’
z
F’
Three level implementation
F
(c) F’=x’y+xy’+z
IMPLEMENTATION OF THE FUNCTION WITH NAND GATES
F(x,y,z) = Σ(0,6)
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21. NOR Implementation
F
(C+D)
A
B
C
D
E
(A+B)
Simplified
Boolean function
F=(A+B)(C+D)E
F
A
B
C
D
E’
Convert OR gates
with at least two
inputs to NOR
gates at first level
(a)
(b)
THREE WAYS TO IMPLEMENT
F=(A+B)(C+D)E
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22. A
B
C
D
E
(C)
Draw a single
NOR gate at level
2 & a NOR gate
for a term with
single literal.
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23. Example
F(n, y, z) = {(0, 6)
F = x’y’z’ + xyz’
F’ = x’y + xy’ + z
x
y’
x’
y
z’
F
(a) F = (x + y’) (x’ + y)z’
x
y
z
x’
y’
z
F
F’
(b) F’ = (x + y + z) (x’ + y’ + z)
IMPLEMENTATION WITH NOR GATES
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24. y
0100
0100
01X0
X11X
01
11
10
YZ
00 01 11 10
w
x
z
wx
00
F(w,x,y,z)=
Σ(1,3,7,11,15)
D(w,n,y,z)=Σ(0,2,5)
Combing 1’s and X’s
F=w’z+yz
or
F=w’x’+yz
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25. YZ
y
0100
0100
01X0
X11X
01
11
10
00 01 11 10
w
x
z
wx
00
Combing 0’s and X’s
F’=z’+wy’
F=z(w’+y)
EXAMPLES WITH DON’T CARE CONDITION
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26. DETERMINE THE PRIME-IMPLICANTS OF THE FUNCTION
F(w,x,y,z)=∑(1,4,6,7,8,9,10,11,15)
(a) (b) (c)
0001 1
0100 4
1000 8
0110 6
2 1 9
1010 10
0111 7
2 11 11
1111 15
1, 9 (8)*
4, 6 (2)*
8, 9 (1)
8, 10 (2)
6, 7 (1)*
9, 11 (2)
10,11 (1)
7, 15 (8)*
11,15 (4)*
8, 9, 10, 11, (1,2)*
8, 9, 10, 11, (1,2)*
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27. PRIME-IMPLICANTS
Decimal Binary
w x y z
Term
1,9 (8)
4,6 (2)
6,7 (1)
7,15 (8)
11,15 (4)
8,9,10,11 (1,2)
- 0 0 1
0 1 - 0
0 1 1 -
- 1 1 1
1 - 1 1
1 0 - -
X’ y’ z
W’ x z’
w’ x y
X y z
w y z
W x’
F=x’y’z+w’xz’+w’xy+xyz+wyz+wx’
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28. (i) Selection of
PRIME-IMPLICANTS
1 4 6 7 8 9 10 11 15
x’y’z 1,9 X X
w’xz’ 4,6 X X
w’xy 6,7 X X
xyz 7,15 X X
wyz 11,15 X X
wx’ 8,9,10,11 X X X X
F=x’y’z+w’xz’+wx’+xyz
Essential Prime-implicants
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29. 1111
1
111
1
01
11
10
YZ
00 01 11 10
w
x
z
wx
00
F(w,x,y,z)=∑(1,4,6,7,8,9,10,11,15)
y
MAP FOR THE FUNCTION OF
F = x’y’z+w’xz’+xyz+wx’
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30. THE TABULATION METHOD (Q-M Mehthod)
Example:
F(w,x,y,z) = ∑(0,1,2,8,10, 11,14,15)
(i) DETERMINATION OF PRIME-IMPLICANTS
(a) (b) (c)
wxyz wxyz wxyz
0 0000√ 0,1 000-*
0,2 00-0√
0,2,8,10 -0-0*
0,8,2,10 -0-0*
1 0001√ 0,8 -000√
2 0010√
10,11,14,15 1-1-*
10,11,14,15 1-1-*
8 1000 √ 2,10 -010√
8,10 10-0√
10 1010√
10,11 101-√
11 1011√ 10,14 1-10√
14 1110√
11,15 1-11√
15 11111√ 14,15 111-√
Prime-implicants
F=w’x’y’+x’z’+wy
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31. (i) Selection of
PRIME-IMPLICANTS
0 1 2 8 10 11 14 15
w’x’y’ 0,1 X X
x;z’ 0,2,8,10 X X X X
wy 10,11,14,15 X X X X
Essential Prime-implicants : F=w’x’y’+x’z’+wy
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32. 111
11
11 1
00 01 11 10
Y
10
11
01
00
wx
yz
W
X
F= w’x’y’ + x’z’+wy
MAP FOR THE FUNCTION
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33. RULES FOR NAND and NOR IMPLEMENTATION
Case
Function
to
Simplify
Standrad
From to
Use
How to
drive
Implement
with
Number
of Levels
to F
(a)
(b)
(c)
(d)
F
F
F
F
Sum of
Products
Sum of
Products
Product
of Sums
Product
of Sums
Combine 1,s
in map
Combine 0, s
In map
Complement
F in (b)
Complement
F in (a)
NAND
NAND
NOR
NOR
2
3
2
3
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34. A
B
C
D
F
F = (AB + CD)’
F = (A B)’ . (C D)’
(a) Wired – AND in open Collector TTL NAND
gates
C
A
B
D
F
F = [(A + B ) ( C + D)]’
F = (A + B)’ + (C + D)’
(b) Wired – OR in ECL gates
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35. A
B
C
D
E
A
B
C
D
E
F F
(AND – OR - INVERT) OR – AND - INVERT
WIRED LOGIC
A
B
C
D
E
F
(a) AND - NOR (b) AND - NOR
(c) NAND - AND
AND – OR – INVERT CIRCUITS F = (AB + CD + E)’
(Non degenerate form)
AND - - NOR & NAND – AND are equivalent forms
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36. NONDEGENERATE FORMS
COMMON GATE : AND, OR, NAND, NOR
IF AT LEVEL 1 : ONE TYPE OF GATE
AT LEVEL 2 : ONE TYPE OF GATE
* POSSIBLE COMBINATIONS = 16
8 OF THESE COMBINATIONS ARE SAID TO BE DEGENERATE FORMS
BECAUSE THEY DEGENERATE TO A SINGLE OPERATION.
FOR EXAMPLE :
NAND – OR AND – AND OPERATION
NAND – NOR OR – OR OPERATION
NOR – AND AND – NAND OPERATION
NOR – NAND OR – NOR OPERATION
8 OF THESE COMBINATION S ARE SAID TO BE NON – DEGENERATE
FORMS BECAUSE THEY PRODUCE AN IMPLEMENTATION IN SUM OF PRODUCTS OR
PRODUCT OF SUMS.
FOR EXAMPLE :
AND – OR OR – AND
NAND – NAND NOR – NOR
NOR – OR NAND – AND
OR – NAND AND - NOR
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37. A
B
C
D
E
F
(a) OR - NAND
A
B
C
D
E
F
(b) OR - NAND
CONTED
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38. A
B
C
D
E
F
(C) NOR - OR
OR – AND – INVERT F = [(A + B ) (C + D ) E]’
(Non – degenerate form)
OR – NAND & NOR – OR ARE EQUIVALENT FORMSFaaDoOEngineers.com

39. X’
y
X
y’
z
F
X’
y
X
y’
z
F
Example
AND - NOR NAND - AND
F = x’ y + x y’ + z
(a) F = (x’ y + x y’ + z)’
1 0 0 0
0 0 0 1
0
1
00 01 11 10
F’ = x’ y + x y’ + z
F = x’ y z’ + x y z’
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40. x
y
z
z
x’
y’
F
x
y
z
z
x’
y’
F
OR - NAND NAND - OR
F = x’ y’ z’ + x y z’
F = (x + y + z) (x’ + y’ +z)
(b) F = [(x + y + z) (x’ + y’ + z)]’
TWO – LEVEL IMPLEMENTATION
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