k-map (2b)upload.wikimedia.org/wikiversity/en/1/13/dd2.a.3.kmap.20130917.pdfsep 17, 2013 · k-map...
TRANSCRIPT
![Page 1: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/1.jpg)
Young Won Lim9/18/13
K-Map (2B)
![Page 2: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/2.jpg)
Young Won Lim9/18/13
Copyright (c) 2011-2013 Young W. Lim.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".
Please send corrections (or suggestions) to [email protected].
This document was produced by using OpenOffice and Octave.
![Page 3: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/3.jpg)
K-Map (2B) 3 Young Won Lim9/18/13
K-Map 4 variables (1)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
1011010000
01
10
11
index minterms
![Page 4: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/4.jpg)
K-Map (2B) 4 Young Won Lim9/18/13
K-Map 4 variables (2)
0 0 1 1
0 0 1 0
0 0 0 1
0 0 0 0
0 1 1 1
0 1 1 0
0 1 0 1
0 1 0 0
1 0 1 1
1 0 1 0
1 0 0 1
1 0 0 0
1 1 1 1
1 1 1 0
1 1 0 1
1 1 0 0
3
2
0
1
7
6
4
5
11
10
8
9
15
14
12
131000 1001 1011 1010
1100 1101 1111 1110
0100 0101 0111 0110
0000 0001 0011 0010
2310
6754
101198
14151312
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
x y z w
x y z w
x y z w
x y zw
z=1z=0
w=1w=0 w=0
x=1
x=0
y=1
y=0
y=0
index minterms
![Page 5: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/5.jpg)
K-Map (2B) 5 Young Won Lim9/18/13
K-Map 3 variables (1)
2310
6754
y=1y=0z=1z=0 z=0
x=0
x=1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
![Page 6: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/6.jpg)
K-Map (2B) 6 Young Won Lim9/18/13
K-Map 3 variables (2)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y
x y
x y
x y
x y z + x y z = x y ( z+z) = x y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x y x y
x y x y
index minterms
x y z
21
a group of 2 minterms
![Page 7: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/7.jpg)
K-Map (2B) 7 Young Won Lim9/18/13
K-Map 3 variables (3)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x z
x z
x z
x z
x y z + x y z = x z( y+y) = x z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
x z x z
x z x z
index minterms
x y z
21
a group of 2 minterms
![Page 8: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/8.jpg)
K-Map (2B) 8 Young Won Lim9/18/13
K-Map 3 variables (4)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y z
y z
y z
y z
x y z + x y z = y z ( x+x ) = y z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y z y z y z y z
index minterms
x y z
21
a group of 2 minterms
![Page 9: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/9.jpg)
K-Map (2B) 9 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x
x
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 x
x
index minterms
x y z
22
a group of 4 minterms
![Page 10: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/10.jpg)
K-Map (2B) 10 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
y
y
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1
y y
index minterms
x y z
22
a group of 4 minterms
![Page 11: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/11.jpg)
K-Map (2B) 11 Young Won Lim9/18/13
K-Map 3 variables (5)
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
z
z
2310
6754
y=1y=0
z=1z=0 z=0
x=0
x=1
10110100
0
1 z z
index minterms
x y z
22
a group of 4 minterms
![Page 12: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/12.jpg)
K-Map (2B) 12 Young Won Lim9/18/13
K-Map, minterms, and Maxterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
3
2
0
1
7
6
4
5
x y z
x y z
x y z
x y z
x y z
x y z
x y z
x y z
2310
6754
10110100
0
1
x y zindex minterms
m0 m1 m2m3
m4 m5m6m7
2310
6754
10110100
0
1
M 0 M 1 M 2M 3
M 4 M 5 M 6M 7
x y z
![Page 13: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/13.jpg)
K-Map (2B) 13 Young Won Lim9/18/13
1
0
1
0
0
0
0
1
Boolean Function with minterms
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F
1
11
1F
2310
6754
10110100
0
1
x y z
1 1
1
F = x z + x y z
![Page 14: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/14.jpg)
K-Map (2B) 14 Young Won Lim9/18/13
Boolean Function with Maxterms
1
0
1
0
0
0
0
1
0 1 1
0 1 0
0 0 1
0 0 0
1 1 1
1 1 0
1 0 1
1 0 0
x y z
3
2
0
1
7
6
4
5
F0
0
0
0
0
F
0
2310
6754
10110100
0
1
x y z
0 0
0 0 0
F = ( x+ z)( y+z)(x+z)
x+ z y+z
x+z
![Page 15: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/15.jpg)
K-Map (2B) 15 Young Won Lim9/18/13
Implicant
P ⇒ Fimplies
F takes the value 1, whenever P equals 1
assuming P is a product term in a sum of products
f (x , y , z ,w) = xy + yz + w
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
(x=1, y=1)
(x=1, y=1, z=1)
(x=1, y=1, z=1, w=1)
general, reduced
![Page 16: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/16.jpg)
K-Map (2B) 16 Young Won Lim9/18/13
Prime Implicant
Prime Implicant: An implicant that is minimal
The removal of any literal from P results in a non-implicant for F
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
(x=1, y=1)
(x=1, y=1, z=1)
(x=1, y=1, z=1, w=1)
general, reducedPrime Implicant
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
A process of expanding terms
general, reduced
![Page 17: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/17.jpg)
K-Map (2B) 17 Young Won Lim9/18/13
z=1
y=1
x=1
Expanding Terms
1 1
1 1 1 1
1 1 1
1 1
w=1
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
xyzw ⇒ f
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Prime Implicant Prime Implicant Prime Implicant
![Page 18: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/18.jpg)
K-Map (2B) 18 Young Won Lim9/18/13
z=1
y=1
x=1
Prime Implicant Example
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Prime Implicant Prime Implicant Prime Implicant
Prime Implicant Prime Implicant Prime Implicant
![Page 19: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/19.jpg)
K-Map (2B) 19 Young Won Lim9/18/13
Essential Prime Implicant
Essential Prime Implicant: prime implicants
that cover an output of the function that no combination of other implicants is able to cover
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
z=1
y=1
x=1
1 1
1 1 1 1
1 1 1
1 1
w=1
Essential Prime Implicant Essential Prime Implicant Essential Prime Implicant
![Page 20: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/20.jpg)
K-Map (2B) 20 Young Won Lim9/18/13
Implicant
P ⇒ Fimplies
F takes the value 1, whenever P equals 1
assuming P is a product term in a sum of products
f (x , y , z ,w) = xy + yz + w
xy f (x , y , z ,w) = xy + yz + w⇒
xyz f (x , y , z ,w) = xy + yz + w⇒
xyzw f (x , y , z ,w) = xy + yz + w⇒
implicants
General,Reduced
![Page 21: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/21.jpg)
K-Map (2B) 21 Young Won Lim9/18/13
2's Complement
![Page 22: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/22.jpg)
K-Map (2B) 22 Young Won Lim9/18/13
Decimal to Binary (1)
![Page 23: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/23.jpg)
K-Map (2B) 23 Young Won Lim9/18/13
Decimal to Binary (2)
![Page 24: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/24.jpg)
K-Map (2B) 24 Young Won Lim9/18/13
Laplace Equation
Decimal
![Page 25: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/25.jpg)
K-Map (2B) 25 Young Won Lim9/18/13
Laplace Equation
![Page 26: K-Map (2B)upload.wikimedia.org/wikiversity/en/1/13/DD2.A.3.KMap.20130917.pdfSep 17, 2013 · K-Map (2B) 3 Young Won Lim 9/18/13 K-Map 4 variables (1) 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0](https://reader033.vdocuments.us/reader033/viewer/2022043003/5f82f12a920697063e679adf/html5/thumbnails/26.jpg)
K-Map (2B) 26 Young Won Lim9/18/13
Laplace Equation
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Young Won Lim9/18/13
References
[1] http://en.wikipedia.org/[2] http://planetmath.org/[3] M.L. Boas, “Mathematical Methods in the Physical Sciences”