k map simplification

Simplification of Boolean functions by Karnaugh Map and Tabulation

Method

Presented ByC.Ramesh

Assistant Professor/ECEKIT-kalaignarkarunanidhi Institute Of Technology,

Ciombatore.

Introduction• Although the truth table representation of a function is unique,

when expressed algebraically, it can appear in many different form.

• Boolean functions may be simplified by algebraic means.

• How ever, this procedure of minimization is difficult because it takes specific rules to predict each succeeding step in the manipulative process.

• The map method provides a simple straight forward procedure for minimizing Boolean functions.

Karnaugh Map (K-Map)• A K-Map is a graphical representation of a truth table that can be

used to reduce a logic circuit to its simplest terms.• The Karnaugh map uses a rectangle divided into rows and

columns.• Any product term in the expression to be simplified can be

represented as the intersection of a row and a column.• The rows and columns are labeled with each term in the expression

and its complement. • The labels must be arranged so that each horizontal or vertical

move changes the state of one and only one variable.

Two variable K-Map

x’y’ x’y

xy’ xy

x

y

0

0

1

1

m0 m1

m2 m3

x

y

0

0

1

1

Three variable K-Map

x’y’z’ x’y’z x’yz x’yz’

xy’z’ xy'z xyz xyz’

m0 m1 m3 m2

m4 m5 m7 m6

x

x

yz

yz

0

1

0

1

00 01 11 10

00 01 11 10

Four variable K-Map

m0 m1 m3 m2

m4 m5 m7 m6

m12 m13 m15 m14

m8 m9 m11 m10

00 01 11 10

00

01

11

10

wx

yz

Four variable K-Map

w’x’y’z’ w’x’y’z w’x’yz w’x’yz’

w’xy’z’ w’xy’z w’xyz w’xyz’

wxy’z’ wxy’z wxyz wxyz’

wx’ y’z’ wx’ y’z wx’yz wx’yz’

00 01 11 10

00

01

11

10

wx

yz

Simplification Using the K-Map

• Look for adjacent squares.• Adjacent squares are those squares where only one variable

changes as one moves from a square to another.• Group adjacent squares in powers of two, i.e. pairs, quads,

groups of eight, groups of 16, etc.• Principle 1. “The more, the merrier.” Hence, a quad is better

than a pair.• Principle 2. Share group elements only when necessary to form

another or bigger group.

f(A,B) = ∑m(1,2,3) f(A,B) = ∑m(0,1)

f(x1,x2,x3) = ∑m(0,1,2,3,6)

f(x1,x2,x3) = ∑m(2,3,4,6)

f(x1,x2,x3,x4,x5) = ∑m(6,7,8,9,12,13,18,22,23,24,25,28,29)

f(a,b,c,d,e) = ∑m(0,2,6,8,9,10,11,18,22,26,24,25,27,30)

POS Simplification

00 01 11 10

00 1 1 0 1

01 0 1 0 0

11 0 0 0 0

10 1 1 0 1

Simplify the function f(A,B,C,D)=Σ(0,1,2,5,8,9,10) in sum of products and product of sums.

00 01 11 10

00 1 1 1

01 1

11

10 1 1 1

POS Simplification

00 01 11 10

00 0

01 0 0 0

11 0 0 0 0

10 0

f(A,B,C,D)=(A’+B’)(C’+D’)(B’+D)

f‘(A,B,C,D)=AB+CD+BD’

Minimize the following Boolean functions by K-Map

THANK YOU