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

Boolean 3.2 K-Mapping

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.

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 AB C CD 3-Variable K-Map 4-Variable K-Map

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

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.

Boolean 3.7 Filling in a K-Map: 3-Variable 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

Boolean 3.8 Filling in a K-Map: 3-Variable 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 = Animated

Boolean 3.9 Exercise AB C Fill in the K-Map for the following Boolean equation: AB’C+ABC + ABC’ +AB’C’

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.

Boolean 3.11 K-Mapping examples: 3-variable AB C AB C AB C AB C Circle of 4 Circle of 2 Circle of 4 (wraps around to the other side)

Boolean 3.12 K-Mapping examples: 4-variable AB CD AB CD AB CD AB CD Circle of 4 (wraps to other side) Circle of 4 (wraps to other sides) Circle of 2 Circle of 4

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.

Boolean variable solution: Step AB C Values encompassed by the circle

Boolean variable solution: Step AB C Opposites cancel After cancellation, the remainder is A=0 Answer: A = output

Boolean 3.16 Example: 3-variable solutions AB C AB C AB C AB C A’ C A’C’ B’

Boolean 3.17 Example: 4-variable solutions AB CD AB CD AB CD AB CD BD’ B’D’ A’BD A’B’

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.

Boolean 3.19 Example: multiple circles AB C A’C’ + AC + AB’ Sometimes there are several different options for circles.

Boolean 3.20 Exercise 2: Determine the Boolean equations AB CD AB CD AB CD AB CD

Boolean 3.21 Truth Table direct to K-Map example INPUTOUTPUT ABCW ◊Values from a Truth Table can be implemented directly into a K-Map AB C

Boolean 3.22 Exercise 3: Simplify using a K-Map INPUTOUTPUT ABCW ◊Instructions: Determine the Boolean equation and simplify using a K- Map AB C Boolean

Boolean 3.23 Exercise 4: Simplify using a K-Map INPUTOUTPUT ABCW

Boolean 3.24 Exercise 5: Simplify using K-Map INPUTOUTPUT ABCDX

Boolean 3.25 ©Paul R. Godin prgodin gmail.com END