1. Boolean 3.1 Boolean Logic 3 ©Paul Godin Created September 2007 Last edit Sept 2009

2. Boolean 3.2 K-Mapping

3. Boolean 3.3 K-Mapping ◊Karnaugh (pronounced “karno”) developed a visual technique for simplifying Boolean equations. ◊K-Mapping relies on pattern recognition. ◊K-Maps are considered easier to use than straight Boolean reduction. ◊K-Maps are in a grid configuration and can be easily used to resolve up to 4-variable problems.

4. Boolean 3.4 K-Mapping ◊K-Map configurations use a Gray Code count as follows: 00, 01, 11, 10 ◊Remember Gray Codes only changes one bit at a time. 01 00 01 11 10 00011110 00 01 11 10 AB C CD 3-Variable K-Map 4-Variable K-Map

5. Boolean 3.5 K-Mapping – Alternate Labeling CC’ A’B’ A’B AB AB’ C’D’C’DCDCD’ A’B’ A’B AB AB’ 3-Variable K-Map 4-Variable K-Map

6. Boolean 3.6 Filling in a K-Map ◊The AND statements are entered directly into the K-Map. ◊The K-Map technique lends itself well to S.O.P. form (where AND statements are OR’d together, such as: ABC+AB’C+AB’C’) ◊K-Maps work well with Truth Tables.

7. Boolean 3.7 Filling in a K-Map: 3-Variable 01 00 01 11 10 AB C Each position in the table represents a Boolean value. A “0” signifies a NOT. A’B’C’ A’B’C A’BC’ A’BC ABC’ ABC AB’C’ AB’C

8. Boolean 3.8 Filling in a K-Map: 3-Variable 01 00 01 11 10 AB C Fill in the K-Map for the following Boolean equation: ABC’+ABC + AB’C’ +AB’C Each “true” value = 1 Any “false” value = 0 11 1 1 0 0 0 0 Animated

9. Boolean 3.9 Exercise 1 01 00 01 11 10 AB C Fill in the K-Map for the following Boolean equation: AB’C+ABC + ABC’ +AB’C’

10. Boolean 3.10 K-Mapping Rules ◊The objective is to circle all of the 1’s using a basic set of rules: ◊Include only 1’s (or x’s) in the circle. No zeros. ◊Make the circle a big as possible. ◊The circle must be either 1, 2, 4, 8 or 16 variables in size. ◊The circle can only be horizontal or vertical, not diagonal. ◊The K-Map wraps around. The corners are connected as are opposite sides. ◊Make the fewest circles possible. ◊Values in a circle are ANDed together; extra circles are ORed together.

11. Boolean 3.11 K-Mapping examples: 3-variable 01 001 1 011 1 110 0 100 0 AB C 01 000 1 010 1 110 1 100 1 AB C 01 001 0 011 0 110 0 100 0 AB C 01 001 1 010 0 110 0 101 1 AB C Circle of 4 Circle of 2 Circle of 4 (wraps around to the other side)

12. Boolean 3.12 K-Mapping examples: 4-variable 00011110 001111 010000 110000 100000 AB CD 00011110 000000 010110 110000 100000 AB CD 00011110 000000 011001 111001 100000 AB CD 00011110 001001 010000 110000 101001 AB CD Circle of 4 (wraps to other side) Circle of 4 (wraps to other sides) Circle of 2 Circle of 4

13. Boolean 3.13 Reading a K-Map ◊Once the circles have been done, the area the circle encompasses is included as part of the equation. ◊Opposite values cancel. This is the simplification process.

14. Boolean 3.14 3-variable solution: Step 1 01 001 1 011 1 110 0 100 0 AB C Values encompassed by the circle

15. Boolean 3.15 3-variable solution: Step 2 01 01 1 01011 1 110 0 100 0 AB C Opposites cancel After cancellation, the remainder is A=0 Answer: A = output

16. Boolean 3.16 Example: 3-variable solutions 01 001 1 011 1 110 0 100 0 AB C 01 000 1 010 1 110 1 100 1 AB C 01 001 0 011 0 110 0 100 0 AB C 01 001 1 010 0 110 0 101 1 AB C A’ C A’C’ B’

17. Boolean 3.17 Example: 4-variable solutions 00011110 001111 010000 110000 100000 AB CD 00011110 000000 010110 110000 100000 AB CD 00011110 000000 011001 111001 100000 AB CD 00011110 001001 010000 110000 101001 AB CD BD’ B’D’ A’BD A’B’

18. Boolean 3.18 K-Map: multiple circles ◊More than one circle may be required to circle all of the 1’s. ◊Circles may overlap. ◊When multiple circles are encountered, the solutions for each circle are OR’d together.

19. Boolean 3.19 Example: multiple circles 01 0010 0110 1101 1011 AB C A’C’ + AC + AB’ Sometimes there are several different options for circles.

20. Boolean 3.20 Exercise 2: Determine the Boolean equations 00011110 000110 010000 110011 100000 AB CD 00011110 000000 011111 110011 100011 AB CD 00011110 001001 011100 110000 101001 AB CD 00011110 001001 011111 111001 101001 AB CD

21. Boolean 3.21 Truth Table direct to K-Map example INPUTOUTPUT ABCW 0001 0011 0101 0111 1000 1010 1100 1110 ◊Values from a Truth Table can be implemented directly into a K-Map. 01 00 11 01 11 11 00 10 00 AB C

22. Boolean 3.22 Exercise 3: Simplify using a K-Map INPUTOUTPUT ABCW 0000 0011 0100 0111 1001 1011 1101 1111 ◊Instructions: Determine the Boolean equation and simplify using a K- Map. 01 00 01 11 10 AB C Boolean

23. Boolean 3.23 Exercise 4: Simplify using a K-Map INPUTOUTPUT ABCW 0000 0011 0100 0111 1001 1010 1101 1110

24. Boolean 3.24 Exercise 5: Simplify using K-Map INPUTOUTPUT ABCDX 00001 00011 00101 00111 01000 01010 01100 01110 10001 10011 10101 10111 11000 11010 11101 11111

25. Boolean 3.25 ©Paul R. Godin prgodin ° @ gmail.com END