Karnaugh map minimization, Quine mcCluskey method- Don;t care Conditions

Karnaugh Maps(K- Map)

—  A Karnaugh map is a graphical representation of the logic system.

—  It can be drawn directly from either minterm (sum-of-products) or maxterm (product-of-sums) Boolean expressions

—  It is desired to have a minimized sum-of-products or a minimized product-of-sums expression.

— Types of  K- Map

— Two-variable Karnaugh map

Construction of a Karnaugh Map(K- Map)

—  In the case of a maxterm Karnaugh map,

—  ‘1’ is placed in all those squares for which the output is ‘0’,

—  ‘0’ is placed for input entries  corresponding to a ‘1’ output. Again, 0s are omitted for simplicity,

—  an ‘X’ is placed in squares  corresponding to ‘don’t care’ conditions.

Two-variable Karnaugh map.

Three-variable Karnaugh map.

Four-variable K- map.

Five-variable K- map.

six-variable K- map.

—  For  minterms (sum of product )table  .

—  For  maxterms (product of sum )table  .

Three-variable K map.

Grouping of adjacent one pair.

—  If we consider the minterm of K map. Then

—  Output of the K map is Y = B

— Three  variable K map 

  • Output of the K map is Y =       
Output of the K map is Y =   
Output of the K map is Y =       
Output of the K map is Y =         
Output of the K map is Y =         

—  Output of the K map is Y =  C

—  Output of the K map is Y =  B

Output of the K map is Y = 

Four-variable K- map.

A B C D

 F

0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1
10 1 0 1 0
11 1 0 1 1
12 1 1 0 0
13 1 1 0 1
14 1 1 1 0
15 1 1 1 1

 

— Example:

Minimize the flowwing function using K – map. Right truth table and Drow the logic ckt diagram,

— Example:

Minimize the flowing function using K – map. Right truth table and Drown the logic ckt diagram,

Five-variable K- map.

— Example:

Minimize the flowing function using K – map. Right truth table and Drown the logic ckt diagram,

—  Example:

Minimize the flowing function using K – map. Right truth table and Drown the logic ckt diagram,

DON’t Care condition

—  In few logic circuits, output is not given or specified for certain combination of input.

—  In such case output level is not defined, it can ‘high’ or ‘low’ either ‘1’ or ‘0’

—  These output are denoted are ‘d’ or ‘x’or ‘Φ

Example:

Minimize the flowing function using K – map. Right truth table and implementing  the circuit .

— Example:

Minimize the the circuit using NAND  gate only. flowing function using K – map. Right truth table and implementing

Minimization of product of Sum from using K map