The document discusses Karnaugh maps, which are a graphical technique for simplifying boolean functions. A K-map is a diagram with squares that each represent minterms or maxterms. Variables are represented along rows and columns. Groups of 1s can be combined according to grouping rules to simplify boolean expressions. The example shows a 2-variable K-map used to minimize the boolean expression XY' + X'Y + X'Y' to X' + Y'. K-maps allow boolean functions to be reduced more easily than boolean algebra.

1. K-MAP

2. Minterm Maxterm
XYZ Term Designation Term Designation
000 X’Y’Z’ M0 X+Y+Z M0
001 X’Y’Z M1 X+Y+Z’ M1
010 X’YZ’ M2 X+Y’+Z M2
011 X’YZ M3 X+Y’+Z’ M3
100 XY’Z’ M4 X’+Y+Z M4
101 XY’Z M5 X’+Y+Z’ M5
110 XYZ’ M6 X’+Y’+Z M6
111 XYZ M7 X’+Y’+Z’ M7

3.  Q1. find the sum of product and product of sum:
f= Σ(1,3,7) (S.O.P)
(X’Y’ Z) + (X’Y Z) + (XY Z)
f=λ (1,3,7) (P.O.S)
(X +Y + Z’) (X +Y’ + Z’) (X’ +Y’ + Z’)

4.  K-Map method is a graphical technique for simplify
boolean function. It is a 2-D representation of a
Truth-Table.
 Karnaugh maps reduce logic functions more quickly
and easily compared to Boolean algebra. By reduce
we mean simplify, reducing the number of gates
and inputs.
 A K-Map is a diagram consisting of squares and
each square of the map represents minterm and
maxterm.

5.  It has 4 minterms map consist of 4 square. One for
each minterm.
 0 and 1 marked for each row and each column
designate the values of variable x and y repectively
 2n = 22= 4 cells.

6.  Note: (Here generally we are taking sum of product
expression rules)
 Groups may not contain Zero( means when we use
Boolean expression with sum of product form then
we use 1 otherwise we use 0)
 We can group 1,2,4,8 (2n cells)
 Each group should be as large as possible.
 Cells containing 1 must be grouped.
 Groups may overlap.
 Opposite grouping and corner grouping is allowed.
 There should be as few group as possible.

7.  Simplify the given 2-variable Boolean equation by using K-map.
 F = XY’ + X’Y + X’Y’
 We put 1 at the output terms given in equation.
 In this K-map, we can create 2 groups by following the rules for grouping, one is
by combining (X’,Y) and (X’,Y’) terms and the other is by combining (X,Y’) and (X’,
Y’) terms. Here the lower right cell is used in both groups.
After grouping the variables, the next step is determining the minimized
expression.
 By reducing each group, we obtain a conjunction of the minimized expression
such as by taking out the common terms from two groups, i.e. X’ andY’. So the
reduced equation will be X’ +Y’.

8. Ex1: AB+AB’+A’B
 So, the minimize expresion is:
 F=A+B
Ex2 : F = ∑ (m0, m1, m2) = A̅B̅ +A̅B +AB̅
Q1: y=a’b+ab+ab’
Q2: x’y+xy’