K-map method
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Transcript of K-map method
![Page 1: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/1.jpg)
G H PATEL COLLEGE OF ENGINEERING AND TECHNOLOGYDEPARTMENT OF INFORMATION TECHNOLOGY
Subject : 2131004 (Digital Electronics)
Preparad By:Harekrushna Patel (130110116035)
K-map Method
![Page 2: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/2.jpg)
Contents
• Introduction• Two variable maps• Three variable maps• Four variable maps• Five variable maps• Six variable maps
![Page 3: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/3.jpg)
Introduction
• The map method provides a simple straight forward procedure for minimizing Boolean functions.
• This method may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram.
• The map method, first proposed by Veitch (1) and slightly modify by Karnaugh (2), is also known as the ‘Veitch diagram’ or the ‘Karnaugh map’.
![Page 4: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/4.jpg)
Cont.
Minterm
• Standard Product Term• For n – variable function → 2n minterm• Sum of all minterms = 1 i.e. ∑mi = 1
![Page 5: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/5.jpg)
Cont.
Maxterm
• Standard Sum Term• For n – variable function → 2n maxterm• Product of all maxterms = 1 i.e. ∏Mj = 1
![Page 6: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/6.jpg)
Cont.
• Forms of Boolean function:– Sum of Product(SOP) form– Product of Sum(POS) form
![Page 7: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/7.jpg)
Cont.
• SOP Form:– AND - OR Logic or NAND - NAND Logic
![Page 8: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/8.jpg)
Cont.
• POS Form:– OR - AND Logic or NOR - NOR Logic
![Page 9: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/9.jpg)
Rules
• No zeros allowed.• No diagonals.• Only power of 2 number of cells in each
group.• Groups should be as large as possible.• Every 1 must be in at least one group.• Overlapping allowed.• Wrap around allowed.• Fewest number of groups possible.
![Page 10: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/10.jpg)
Two variable K-map
• There are four minterms for two variables; hence the map consists of four squares, one for each minterm.
• The 0’s and 1’s marked for each row and each column designate the values of variables x and y, respectively.
mo m1
m2 m3
![Page 11: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/11.jpg)
Cont.
mo m1
m2 m3
![Page 12: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/12.jpg)
Cont.
mo m1
m2 m3
• Take two variables x and y
x y
![Page 13: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/13.jpg)
Cont.
mo m1
m2 m3
• Relation between squares & two variables
xy
0
1
0 1
X’
X
y’ y
![Page 14: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/14.jpg)
Cont.
x’y’
• Relation between squares & two variables
xy
0
1
0 1
X’
X
y’ y
![Page 15: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/15.jpg)
Cont.
x’y’ x’y
• Relation between squares & two variables
xy
0
1
0 1
X’
X
y’ y
![Page 16: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/16.jpg)
Cont.
x’y’ x’y
xy’
• Relation between squares & two variables
xy
0
1
0 1
X’
X
y’ y
![Page 17: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/17.jpg)
Cont.
x’y’ x’y
xy’ xy
• Relation between squares & two variables
xy
0
1
0 1
X’
X
y’ y
![Page 18: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/18.jpg)
Example
• Simplify following two Boolean functions:– F1 = xy– F2 = x+y
![Page 19: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/19.jpg)
Cont.
mo m1
m2 m3
• F1 = xy……????
xy
0
1
0 1
X’
X
y’ y
![Page 20: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/20.jpg)
Cont.
0 0
0 1
• F1 = xy
xy
0
1
0 1
X’
X
y’ y
![Page 21: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/21.jpg)
Cont.
mo m1
m2 m3
• F2 = x + y……????
xy
0
1
0 1
X’
X
y’ y
![Page 22: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/22.jpg)
Cont.
0 1
1 1
• F2 = x + y = x’y + xy’ + xy = m1 + m2 + m3
xy
0
1
0 1
X’
X
y’ y
![Page 23: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/23.jpg)
Three variable K-map
• There eight minterms for three binary variables. Therefore, a map consists of eight squares.
m0 m1 m3 m2
m4 m5 m7 m6
![Page 24: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/24.jpg)
Cont.
m0 m1 m3 m2
m4 m5 m7 m6
![Page 25: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/25.jpg)
Cont.
m0 m1 m3 m2
m4 m5 m7 m6
• Take three variables x, y and z
xyz
![Page 26: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/26.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’
x
m0 m1 m3 m2
m4 m5 m7 m6
y’z’ y’z y z y z’
![Page 27: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/27.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’
y’z’ y’z y z y z’
x
![Page 28: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/28.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z
y’z’ y’z y z y z’
x
![Page 29: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/29.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z x’yz
y’z’ y’z y z y z’
x
![Page 30: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/30.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z x’yz x’yz’
y’z’ y’z y z y z’
x
![Page 31: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/31.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z x’yz x’yz’
xy’z’
y’z’ y’z y z y z’
x
![Page 32: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/32.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z x’yz x’yz’
xy’z’ xy’z
y’z’ y’z y z y z’
x
![Page 33: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/33.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’ x’y’z’ x’y’z x’yz x’yz’
xy’z’ xy’z xyz
y’z’ y’z y z y z’
x
![Page 34: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/34.jpg)
Cont.
• Relation between squares & three variables
xyz
0
1
00 01 11 10
x’
y’z’ y’z y z y z’
x’y’z’ x’y’z x’yz x’yz’
xy’z’ xy’z xyz xyz’x
![Page 35: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/35.jpg)
Example
• Simplify the Boolean function:– F = x’yz + xy’z’ + xyz + xyz’
• Ans.:– x’yz = m3
– xy’z’ = m4
– xyz = m7
– xyz’ = m6
![Page 36: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/36.jpg)
Cont.
xyz
0
1
00 01 11 10
x’
x
m0 m1 m3 m2
m4 m5 m7 m6
y’z’ y’z y z y z’
• F = x’yz + x’yz’ + xy’z’ + xy’z
![Page 37: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/37.jpg)
Cont.
xyz
0
1
00 01 11 10
x’
x
0 0 1 0
1 0 1 1
y’z’ y’z y z y z’
• F = x’yz + x’yz’ + xy’z’ + xy’z
![Page 38: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/38.jpg)
Cont.
xyz
0
1
00 01 11 10
x’
x
0 0 1 0
1 0 1 1
y’z’ y’z y z y z’
• Final Ans. F = yz + xz’
![Page 39: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/39.jpg)
Four Variable K-map
• There sixteen minterms for four binary variables. Therefore, a map consists of sixteen squares.
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
![Page 40: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/40.jpg)
Cont.
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Take four variables A,B,C and D
![Page 41: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/41.jpg)
Cont.
A’B’C’D’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 42: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/42.jpg)
Cont.
A’B’C’D’ A’B’C’D
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 43: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/43.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 44: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/44.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 45: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/45.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 46: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/46.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 47: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/47.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 48: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/48.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 49: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/49.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 50: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/50.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 51: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/51.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 52: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/52.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD ABCD’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 53: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/53.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD ABCD’
AB’C’D’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 54: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/54.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD ABCD’
AB’C’D’ AB’C’D
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 55: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/55.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD ABCD’
AB’C’D’ AB’C’D AB’CD
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 56: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/56.jpg)
Cont.
A’B’C’D’ A’B’C’D A’B’CD A’B’CD’
A’BC’D’ A’BC’D A’BCD A’BCD’
ABC’D’ ABC’D ABCD ABCD’
AB’C’D’ AB’C’D AB’CD AB’CD’
00 01 11 10C’D’ C’D C D C D’
00
01
11
10
A’B’
A’B
A B
A B’
ABCD
• Relation between squares & four variables
![Page 57: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/57.jpg)
Example
• Simplify the Boolean function:– F(w, x, y, z) = Σ(1,5,12,13)
![Page 58: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/58.jpg)
Cont.
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
WZ
XY
00 01 11 10
00
01
11
10
F(w, x, y, z) = Σ(1,5,12,13)
![Page 59: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/59.jpg)
Cont.
0 1 0 0
0 1 0 0
1 1 0 0
0 0 0 0
WZ
XY
00 01 11 10
00
01
11
10
F(w, x, y, z) = Σ(1,5,12,13)
Put 1 in place ofm1, m5, m12, m13
![Page 60: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/60.jpg)
Cont.
0 1 0 0
0 1 0 0
1 1 0 0
0 0 0 0
WZ
XY
00 01 11 10
00
01
11
10
F(w, x, y, z) = Σ(1,5,12,13)
Put 1 in place ofm1, m5, m12, m13
Making pairs
![Page 61: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/61.jpg)
Cont.
0 1 0 0
0 1 0 0
1 1 0 0
0 0 0 0
WZ
XY
00 01 11 10
00
01
11
10
F(w, x, y, z) = Σ(1,5,12,13)
Put 1 in place ofm1, m5, m12, m13
Making pairs
Hence the simplifiedExpression isF = WY’Z + W’Y’Z
![Page 62: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/62.jpg)
Five variable K-map
• There thirty two minterms for five binary variables. Therefore, a map consists of thirty two squares.
m16 m17 m19 m18
m20 m21 m23 m22
M28 m29 M31 m30
m24 m25 m27 m26
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
![Page 63: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/63.jpg)
Cont.
m16 m17 m19 m18
m20 m21 m23 m22
m28 m29 m31 m30
m24 m25 m27 m26
m0 m1 m3 m2
m4 m5 m7 m6
m12 m13 m15 m14
m8 m9 m11 m10
ABCD
• Relation between squares & five variables
E
00
01
11
10
00 01 11 100 0 0 0 11 110 1 1 10 001
![Page 64: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/64.jpg)
Cont.
• Example:– Design a circuit of 5 input variables that generates
output 1 if and only if the number of 1’s in the input is prime (i.e., 2, 3 or 5).
• Ans.:– The minterms can easily be found from Karnaugh
Map where addresses of 2,3 or 5 numbers of 1.
![Page 65: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/65.jpg)
Cont.
![Page 66: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/66.jpg)
Cont.
![Page 67: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/67.jpg)
Cont.
• Hence the simplified expression becomes
BC’D’E + A’BC’D + AC’DE’ + AB’C’D + A’B’CE + A’CDE’ + A’BCD + AB’CD’ + ABD’E’ + AB’DE’ + A’B’DE + ABCDE
![Page 68: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/68.jpg)
6 variable K-map
• A 6-variable K-Map will have 26 = 64 cells. A function F which has maximum decimal value of 63, can be defined and simplified by a 6-variable Karnaugh Map.
![Page 69: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/69.jpg)
Cont.
![Page 70: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/70.jpg)
Cont.
• Boolean table for 6 variables is quite big, so we have shown only values, where there is a noticeable change in values which will help us to draw the K-Map.
• A = 0 for decimal values 0 to 31 and A = 1 for 31 to 63.
• B = 0 for decimal values 0 to 15 and 32 to 47. B = 1 for decimal values 16 to 31 and 48 to 63.
![Page 71: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/71.jpg)
Cont.
No. A B C D E F Minterm
m0 0 0 0 0 0 0 A’B’C’D’E’F’
m15 0 0 1 1 1 1 A’B’CDEF
m16 0 1 0 0 0 0 A’BC’D’E’F’
m31 0 1 1 1 1 1 A’BCDEF
m32 1 0 0 0 0 0 AB’C’D’E’F’
m47 1 0 1 1 1 1 AB’CDEF
m48 1 1 0 0 0 0 ABC’D’E’F’
m63 1 1 1 1 1 1 ABCDEF
![Page 72: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/72.jpg)
Cont.
• Example:– F = Σ (0, 2, 4, 8, 10, 13, 15, 16, 18, 20, 23, 24, 26,
32, 34, 40, 41, 42, 45, 47, 48, 50, 56, 57, 58, 60, 61)
• Ans.:– Since, the biggest number is 61, we need to have
6 variables to define this function.
![Page 73: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/73.jpg)
F = Σ (0, 2, 4, 8, 10, 13, 15, 16, 18, 20, 23, 24, 26, 32, 34, 40, 41, 42, 45, 47, 48, 50, 56, 57, 58, 60, 61)
![Page 74: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/74.jpg)
Cont.
• Hence the simplified expression becomesF = D’F’ + ACE’F + B’CDF + A’C'E’F’ + ABCE’ +
A’BC’DEF
![Page 75: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/75.jpg)
Cont.
• Example:– F = Σ (0, 1, 2, 3, 4, 5, 8, 9, 12, 13, 16, 17, 18, 19,
24, 25, 36, 37, 38, 39, 52, 53, 60, 61)
• Ans.:– Since, the biggest number is 61, we need to have
6 variables to define this function.
![Page 76: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/76.jpg)
Cont.
![Page 77: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/77.jpg)
Cont.
• Hence the simplified expression becomesF = A’B'E’ + A’C'D’ + A’D'E’ + AB’C'D + ABCE’
![Page 78: K-map method](https://reader035.fdocuments.net/reader035/viewer/2022081506/55920ead1a28ab057e8b465d/html5/thumbnails/78.jpg)
THANK YOU...
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