E- Lesson: Karnaugh Maps (K-Map )

Subject: Computer Science (083) Section: Boolean Algebra Topic: Minimization of Boolean Expression using K-Map Category: Senior Secondary Class: XII

Instructional Objective The instructional objective of lesson is…… To enable the students to acquire the knowledge of Simplifying boolean expression. To Understand Need of Simplifying boolean expression To implement the knowledge for Simplifying boolean expression

Learning Objective After successfully completion of this module students should be able to, Understand Need of Simplifying expression List of different Methods for minimization. Algebraic Method Karnaugh Map

Pre- Requisites Information Mintern Maxterm Canonical SOP Canonical POS Logic Gates

Importance of K-Map. A K-Map provide a pictorial method of grouping together expression with common factors and therefore eliminating unwanted variables from function. K- Map is extensively used for simplification of boolen expression with up to six variables. The Map consists of 2n squares for n variable function.

Graphical Representation of K-Map

Addressing the Cells Address of K- Maps must be ordered such that the value for adjacent columns vary by a single bit. Like (00,01,11,10). This ordering is known as Gray Code and it is a key factor in the way in which K- Map works. The Example for Mapping are:

Two Variable K-Map

Question ??? Question: Draw the Map for 3 Variable function.

Three-Variable K-Map Where m1,m2……. M7 are shorthand notation for each term.

Four-variable K-Map

Four-variable K-Map

Four-variable K-Map Edges are adjacent Edges are adjacent

Plotting Functions on the K-map SOP Form

using shorthand notation Canonical SOP Form Three Variable Example using shorthand notation

Three-Variable K-Map Example Plot 1’s (minterms) of switching function 1 1 1 1

Four-variable K-Map Example 1 1 1 1 1

Guidelines for grouping Functions In grouping, the No. of groups should be minimum and the size of group should be maximize. First of all make octet by using straight combination and then Roll Mapping method. After finding all Octet, make the Quads by using straight combination and Roll Mapping.

Pair Reduction Rule Simplification Procedure for K – Map Remove the variable which changes its state from complemented to un complemented or vice versa. Pair removes one variable only.

Quad Reduction Rule Remove the two variables which change their state from complemented to un complemented or vice versa. Quad removes two variables.

Octet Reduction Rule Remove three variables which change their state from complemented to un complemented or vice versa.

Map Rolling Map rolling means roll the map considering the map as if its left edges are touching the right edges and top edges are touching bottom edges. While marking the pairs quads and octet, map must be rolled.

Overlapping Groups Overlapping means same 1 can be encircled more than once. Overlapping always leads to simpler expressions.

Redundant Group It is a group whose all 1's are overlapped by other groups. Redundant groups must be removed. Removal of redundant group leads to much simpler expression.

Guidelines for grouping Functions After finding all Quads, make the Pairs by using straight combination and Roll Mapping. Finally remove the completely overlap group and then apply simplification method.

K-map Simplification Procedure To minimize the boolean funtion, remove the variables which vary in rows and columns. ----------------------------------------- No. of. Variable in group Eliminate Variable ----------------------------------------------------------- 2 (Pair) 1 4 (Quad) 2 8 (Octet) 3 -------------------------------------------------------

Example- 1

Example- 2

Example- 3

Example-4

Question-1 Represent the following boolean expression in a K-map and simplify. F = x'yz + x'yz' + xy'z' + xy'z

Solution : Hence the simplified expression is  F = x'y + xy'

Question-2 F = a'bc + ab'c' + abc + abc' Represent the following boolean expression in a K-map and simplify. F = a'bc + ab'c' + abc + abc'

Solution : Hence the simplified expression is  F = bc + ac'

Any Question?