# Karnaugh Maps with Don't Care Conditions

In this tutorial, we will learn about the Karnaugh maps with don't care conditions in Digital Electronics. By Saurabh Gupta Last updated : May 10, 2023

## What are Don't Care Conditions?

Till now, the Boolean expressions which have been discussed by us were completely specified, i.e., for each combination of input variable we have specified a minterm by representing them as 1 in the K-Map. But, there may arise a case when for a given combination of input we may not have a specified output or the input combination may be invalid. The combinations for which we don't have any output expression specified are called **don't care combination**.

### Example

For Example, in 8421 code, input states 1001, 1010, 1011, 1100, 1101, 1110 and 1111 are invalid and the corresponding output is the **don't care**. Similarly, in Excess-3 code, binary input states 0000, 0001, 0010, 1101, 1110 and 1111 are unspecified and are also represented by **don't care**.

## Don't care combinations in the K-Map

These **don't care combinations** in the **K-Map** are denoted by an **X (cross) symbol**.

The **X** is called **"Don't care conditions"**.

## Rules for Karnaugh Maps with Don't Care Conditions

General rules to be followed while minimizing the expressions using K-Map which include **don't care conditions** are as follows,

- After forming the K-Map, fill 1's at the specified positions corresponding to the given minterms. Fill
**X**at the positions where don't care combinations are present. - Now, Encircle the groups in the K-Map. One thing to be kept in mind is, now we can treat
**Don't Care conditions (X)**as**1s**if these help in forming the largest groups. No such group can be encircled whose all the elements are**X**. - If still there are
**1s**left which doesn't get encircled in any of the groups, then these isolated**1s**are encircled individually. - Now, recheck all the encircled groups, and remove any redundancy if present.
- Write the Boolean expression for each encircled group.
- The final minimal expression can be obtained by ORing each Boolean expressions that were obtained from each group.

## Point to remember

While designing K-Map using SOP form, don't care conditions (**X**) are considered as 1, if it helps form the largest group, otherwise it is considered as 0 and are left during encircling. On the contrary, while designing a K-Map using POS form, don't care conditions (**X**) are considered as a 0, if it helps form the largest group, otherwise it is considered as 1 and are left during encircling.

## Karnaugh Maps with Don't Care Conditions Examples

### Example 1

Minimize the given Boolean Expression by using the four-variable K-Map.

F (A, B, C, D) = Σ m (1, 5, 6, 12, 13, 14) + d (2, 4).

### Solution

We will fill **1s** at the appropriate minterm positions and also the don't care positions will be filled with (**X**). Now, we can encircle these **1s** and **X** using the rules we have discussed earlier. We can observe that we can encircle them into three groups. First Group (encircled by black ink) contains four **1s** and the Boolean expression given by it is **B.C**. Second group is the rolling group (encircled by blue ink) which also contains four **1s** gives **B.D**. Third group is the pair of **1s** which gives **A.C.D**

Hence, the resultant simplified Boolean Expression will be:

F (A, B, C, D) = B.C + B.D + A.C.D

### Example 2

Minimize the given Boolean Expression by using the four-variable K-Map.

F (A, B, C, D) = Σ m (1, 5, 6, 12, 13, 14) + d (2, 4).

### Solution

As we have already discussed in our last example, we will fill **1's** at the appropriate minterm positions and **X** at don't care positions, encircling the groups using the rules we have discussed above results into the formation of five groups as seen in the image. The Boolean Expressions given by the groups are **A, C.D, B.C. D, A.B.D and B.C.D respectively**.

ORing these individual Boolean expressions to get the simplified Boolean Expression as:

F (A, B, C, D) = A + C. D + B.C. D + A.B. D + B. C. D

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- Binary Coded Decimal (BCD Code) and Its Addition
- Error-Detecting Code and Parity
- Excess-3 (XS-3) Code and Gray Code
- Logic Gates: Definition, Types, Uses, Symbols, Truth Tables
- Universal Gates: NAND and NOR with Examples
- Duality Principle and Rules for Reduction of Boolean Expressions
- Axioms and Laws of Boolean Algebra
- Realization of Boolean Expressions using Basic Logic Gates
- Realization of Boolean Expressions using only Universal Gates
- Solved Examples on Reduction of Boolean Expression
- Karnaugh Maps (K-Maps): Meaning, Types, Representations
- POS and SOP Form Representations of a Boolean Function
- Min and Max Terms Notation in Boolean Algebra

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