Solution:

What is Karnaugh map?

• A Karnaugh map is similar to a truth table because it presents all of the possible values of input variables and the resulting output for each value.

• Instead of being organized into columns and rows like a truth table, the Karnaugh map is an array of cells in which each cell represents a binary value of the input variables.

• The cells are arranged in a way so that simplification of a given expression is simply a matter of properly grouping the cells.

• Karnaugh maps can be used for expressions with two, three, four, and five variables, but we will discuss only 3-variable and 4-variable situations to illustrate the principles.

• The number of cells in a Karnaugh map, as well as the number of rows in a truth table, is equal to the total number of possible input variable combinations.

• For three variables, the number of cells is 23 = 8. For four variables, the number of cells is 24 = 16.

The 3-Variable Karnaugh Map:

• The 3-variable Karnaugh map is an array of eight cells, as shown in Figure (a). In this case, A, B, and C are used for the variables although other letters could be used.

• Binary values of A and B are along the left side (notice the sequence) and the values of C are across the top.

• The value of a given cell is the binary values of A and B at the left in the same row combined with the value of C at the top in the same column.

• For example, the cell in the upper left corner has a binary value of 000 and the cell in the lower right corner has a binary value of 101.

• Figure (b) shows the standard product terms that are represented by each cell in the Karnaugh map.

The 4-Variable Karnaugh Map:

• The 4-variable Karnaugh map is an array of sixteen cells, as shown in Figure(a).

• Binary values of A and B are along the left side and the values of C and D are across the top.

• The value of a given cell is the binary values of A and B at the left in the same row combined with the binary values of C and D at the top in the same column.

  

• For example, the cell in the upper right corner has a binary value of 0010 and the cell in the lower right corner has a binary value of 1010.

• Figure (b) shows the standard product terms that are represented by each cell in the 4-variable Karnaugh map.