Question

Use a Karnaugh map to find the minimum AND-OR expression for x(a, b, c) with don’t-care conditions:

(c) Σ(1, 3) + d(0, 2, 4, 6)

(e) Σ(1, 7) + d(2, 4)

Answer 1

Solution:----

Don't Care Terms

The “Don’t Care” conditions allow us to replace the empty cell of a K-Map to form a grouping of the variables. While forming groups of cells, we can consider a “Don’t Care” cell as either 1 or 0 or we can simply ignore that cell. Therefore, “Don’t Care” condition can help us to form a larger group of cells.

A Don’t Care cell can be represented by a cross(X) in K-Maps representing a invalid combination.

Significance of “Don’t Care” Conditions:
Don’t Care conditions has the following significance with respect to the digital circuit design:

Simplification:
These conditions denotes the set of inputs which never occurs for a given digital circuits. Thus, they are being used to further simplify the boolean output expression.

1)Lesser number of gates:
Simplification reduces the number of gates to be used for implementing the given expression. Therefore, don’t cares make the digital circuit design more economical.

2)Reduced Power Consumption:
While grouping the terms long with don’t cares reduces switching of the states. This decreases the required memory space which in turn results in less power consumption.

Karnaugh Maps

Karnaugh maps are used to simplify real-world logic requirements so that they can be implemented using a minimum number of physical logic gates. A sum-of-products expression can always be implemented using AND gates feeding into an OR gate, and a product-of-sums expression leads to OR gates feeding an AND gate

So here as per the question we are asked to form the minimum SOP ( Sum of Products ) expression for each of the expressions:----