1. Plotting functions not in canonical form
Plot the function f(a, b, c) = a + bc
ab a ab
c c b
The squares are numbered – derive the canonical form

2. 5-variable K-maps - alternative00 11 10 01 00 11 10 01 1 3 4 5 12 13 15 8 9 2 7 6 14 11 10 18 19 17 22 23 30 31 29 26 27 16 21 20 28 25 24 00 01 11 10 00 01 11 10 1

3. 6-variable K-maps - alternative00 01 40 41 43 44 45 36 37 39 32 33 42 47 46 38 35 34 00 01 11 10 1 3 4 5 12 13 15 8 9 2 7 6 14 18 19 17 22 23 30 31 29 26 27 16 21 20 28 25 24 62 63 61 58 59 54 55 53 50 51 60 57 56 52 49 48 10 11

4. Simplifying functions using K-maps
Why is simplification possible
Logically adjacent minterms are physically adjacent on the K-map
Adjacent minterms can be combined by eliminating the common variable
abc and ābc are adjacent
abc + ābc = bc  variable a eliminated
Done by drawing on the map a ring around the terms that can be combined

5. Simplifying functions using K-maps

6. Simplifying functions using K-maps

7. Simplifying functions using K-maps
Definition of terms
Implicant  product term that can be used to cover minterms
Prime implicant  implicant not covered by any other implicant
Essential prime implicant  a prime implicant that covers at least one minterm not covered by any other prime implicant
Cover  set of prime implicants that cover each minterm of the function
Minimizing a function  finding the minimum cover

8. Simplifying functions using K-maps
Definition of terms
Implicants:

9. Simplifying functions using K-maps
Definition of terms
Prime implicants: only B and AC
Essential prime implicants: B and AC
Cover: { B, AC }

10. Simplifying functions using K-maps
Definition of terms
Implicate  sum term that can be used to cover maxterms (0’s on the K-map)
Prime implicate  implicate not covered by any other implicate
Essential prime implicate  a prime implicate that covers at least one maxterm not covered by any other prime implicate
Cover  set of prime implicates that cover each maxterm of the function

11. Simplifying functions using K-maps
Algorithm 1:
Fast and easy, not optimal

12. Simplifying functions using K-maps
Algorithm 2:
More work than the first
Can give better results, because all prime implicants are considered
Still not optimal

13. Simplifying functions using K-maps
Algorithm 2:
1: Identify all PIs

14. Simplifying functions using K-maps
Algorithm 2:
2: Identify EPIs

15. Simplifying functions using K-maps
Algorithm 2:
3: Select cover

16. The Quine-McCluskey minimization method
Tabular
Systematic
Can handle a large number of variables
Can be used for functions with more than one output

17. The Q-M minimization method

18. The Q-M minimization method

19. The Q-M minimization method

20. The Q-M minimization method
Combine minterms from List 1 into pairs in List 2
Take pairs from adjacent groups only, that differ in 1 bit
Combine entries from List 2 into pairs in List 3

21. The Q-M minimization method

22. The Q-M minimization method

23. The Q-M minimization method

24. The Q-M minimization method