![Page 1: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/1.jpg)
Introduction to Digital Logic
K-Maps
![Page 2: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/2.jpg)
2
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Put in sum of products form
Put in sum of minterms form
Find the maximal p-terms
Find the minterms that only have one maximal p-term
Those are the start of the expression
Remove minterms that have that p-term
Find the maximal p-term that occurs the most times. Pick one if tie.Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm
![Page 3: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/3.jpg)
3
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find all of the minterms
ABCD ABC'D
A'BCD A'BCD' A'BC'D A'BC'D'
ABC'D' A'BC'D'
AB'CD A'B'CD
AB'CD' A'B'CD'
![Page 4: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/4.jpg)
4
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
ABCD ABC'D
A'BCD A'BCD' A'BC'D A'BC'D'
ABC'D'
AB'CD A'B'CD
AB'CD' A'B'CD'
![Page 5: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/5.jpg)
5
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find the maximal p-terms
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
![Page 6: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/6.jpg)
6
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find minterms that only have one maximal p-term
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
![Page 7: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/7.jpg)
7
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Write those as the start to the expression
ABCD (BD, CD) ABC'D (BC', BD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B) A'BC'D (BC', BD, A'B)
A'BC'D' (A'B, BC')
ABC'D' (BC')
AB'CD (CD, B'C) A'B'CD (CD, B'C, A'C)
AB'CD' (B'C) A'B'CD' (B'C, A'C)
BC' + B'C
![Page 8: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/8.jpg)
8
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Remove any minterms that have those maximal p-terms
ABCD (BD, CD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)
BC' + B'C
![Page 9: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/9.jpg)
9
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Find the maximal p-term that occurs the most times. Pick one if tie.
ABCD (BD, CD)
A'BCD (BD, A'B, CD, A'C) A'BCD' (A'C, A'B)
BC' + B'C
![Page 10: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/10.jpg)
10
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
Remove minterms that contain that maximal p-term Pick another maximal p-term, add to the expression, remove minterm
ABCD (BD, CD)
BC' + B'C + A'B
![Page 11: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/11.jpg)
11
Algorithmic Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
BC' + B'C + A'B + CD
![Page 12: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/12.jpg)
12
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
Boxes represent the minterms
Values grouped together
![Page 13: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/13.jpg)
13
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
Find all the minterms and put a 1 in those boxes
![Page 14: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/14.jpg)
14
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1 1 1
1 1
![Page 15: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/15.jpg)
15
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1 1 1
1 1 1
![Page 16: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/16.jpg)
16
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1
1 1 1 1
1 1 1
1
![Page 17: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/17.jpg)
17
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1
1 1 1 1
1 1 1
1
![Page 18: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/18.jpg)
18
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
All the minterms are represented
![Page 19: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/19.jpg)
19
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Group into 8, 4, 2, 1. The bigger the group, the better.
![Page 20: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/20.jpg)
20
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Represented by maximal p-term BC’
![Page 21: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/21.jpg)
21
K-Map Simplification
A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
![Page 22: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/22.jpg)
22
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
![Page 23: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/23.jpg)
23
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
![Page 24: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/24.jpg)
24
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
![Page 25: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/25.jpg)
25
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
All maximal p-terms are represented
![Page 26: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/26.jpg)
26
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
Find mniterms represented by only one p-terms. Find 1s only covered by one box.
![Page 27: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/27.jpg)
27
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
BC’ + B’C
However, all 1s are not covered
![Page 28: Introduction to Digital Logic · 2 Algorithmic Simplification A·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’ Put in sum of products form Put in sum of minterms form Find the](https://reader034.vdocuments.us/reader034/viewer/2022051808/600a85caa25430783313c7b8/html5/thumbnails/28.jpg)
28
K-Map SimplificationA·B·D+A’·B+B·C’·D’+B’·C·D+B’·C·D’
AB00 01
00
01
CD
11 10
11
10
1 1
1 1 1 1
1 1 1
1 1
BC’ + B’C + AB + CD