1. Quine-McClusky Minimization Method Discussion D3.2

2. Quine-McCluskey Method Tabular Representations Prime Implicants Essential Prime Implicants

3. Tabular Representations 111 1 1 WX YZ 00011110 00 01 11 10 111 1 F = XY + W'YZ' + WY'Z + W'X W'X 01-- W'YZ' 0-10 XY -11- WY'Z 1-01

4. Prime Implicants F = XY'Z + X'Z' + X'Y Each product term is an implicant A product term that cannot have any of its variables removed and still imply the logic function is called a prime implicant.

5. Prime Implicants X YZ 00011110 0 1 1 1 11 F = YZ' + X 1 -10 1--

6. Prime Implicants F = YZ' + X X YZ 00011110 0 11 1 111 Minterm X Y Z F 0 O O O 0 10 0 1 0 20 1 0 1 30 1 1 0 4 1 O O 1 51 0 1 1 61 1 0 1 71 1 1 1 -10 1--

7. Finding Prime Implicants 2 0 1 0 4 1 O 0 5 1 0 1 6 1 1 0 7 1 1 1 Step 1Step 2 (2,6) - 1 0 (4,5) 1 0 - (4,6) 1 - 0 (5,7) 1 - 1 (6,7) 1 1 - Step 3 (4,5,6,7) 1 - - (4,6,5,7) 1 - - All unchecked entries are Prime Implicants -10 YZ' 1-- X

8. Prime Implicants F = YZ' + X X YZ 00011110 0 11 1 111 Minterm X Y Z F 0 O O O 0 10 0 1 0 20 1 0 1 31 1 1 0 4 1 O O 1 51 0 1 1 61 1 0 1 71 1 1 1 -10 1--

9. Essential Prime Implicants 11 11 1 WX YZ 00011110 00 01 11 10 11 1 11 Find the essential prime implicants using the Q-M method.

10. Essential Prime Implicants 11 11 1 WX YZ 00011110 00 01 11 10 11 1 11 minterms 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111

11. Finding Prime Implicants 0 0000 1 0001 2 0010 8 1000 3 0011 5 0101 10 1010 7 0111 14 1110 15 1111 Step 1Step 2 (0,1) 000- (0,2) 00-0 (0,8) -000 (1,3) 00-1 (1,5) 0-01 (2,3) 001- (2,10) -010 (8,10) 10-0 (3,7) 0-11 (5,7) 01-1 (10,14) 1-10 (7,15) -111 (14,15) 111- Step 3 (0,1,2,3) 00-- (0,2,1,3) 00-- (0,2,8,10) -0-0 (0,8,2,10) -0-0 (1,3,5,7) 0--1 (1,5,3,7) 0--1 6 Prime Implicants 1-10 -111 111- 00-- -0-0 0--1

12. Find Essential Prime Implicants Prime Implicant Covered minterms 1-10 -111 111- 00-- -0-0 0--1 Minterms 0 1 2 3 5 7 8 10 14 15 10,14 7,15 14,15 0,1,2,3 0,2,8,10 1,3,5,7 XX XX XX XX X X XXXX X XXX *

13. 3 Prime Implicants 11 11 1 WX YZ 00011110 00 01 11 10 11 1 11 0--1 -0-0 111- W'Z WXY X'Z' F = W'Z + WXY + X'Z'

14. 3 Prime Implicants 11 11 1 WX YZ 00011110 00 01 11 10 11 1 11 0--1 -0-0 111- 00-- -111 1-10 W'Z WXY X'Z' F = W'Z + WXY + X'Z'