# Karnaugh Maps: Definition, Types, Essential Terms and Solved Question

The karnaugh Map or K-map is a graphical technique that provides an organized approach for simplifying and operating the Boolean expressions or to transform a truth table to its analogous logic circuit in a simple manner. In this process, the information available in the form of a truth table or SOP (Sum of Product) form or POS (Product of Sum) form is expressed on the K-map.

In many digital circuits and practical questions, we need to determine expressions with minimum variables. We can minimize Boolean expressions of three, four variables very easily using K-map without applying any Boolean algebra theorems.

Through this article on Karnaugh Maps, you will learn about **what is k map, **its implementation through the Sum of Product (SOP) and Product of Sum (POS) form followed by Two-variable, Three-variable, Four-variable, Five-variable K-map and more.

Check out this article on RAM

**Karnaugh Map**

K-map is a table like representation but it provides more information than the truth table. We fill the grid of K-map with 0’s and 1’s then solve it by creating groups.

Although this approach may be applied for any number generally it is used up to six variables, exceeding which it becomes unmanageable. For an n variable K-map, there are 2n cells. In K-map, Gray code is applied for the recognition of cells.

**K Map- Key Terms **

**Minterms and Maxterms**

**Minterms-** It is known as the product term. In the minterm, each uncomplemented term is indicated by ‘1’, and each complemented term is indicated by ‘0’.

**Example-** 011= A’BC

**Maxterms**– It is known as the sum term. In maxterm, each uncomplemented term is indicated by ‘0’ and each complemented term is indicated by ‘1’.

**Example: –** 100= A’+B+C

**Sum of Product (SOP) Form**

In SOP or Sum of Product, form variables are ANDed(i.e multiplied) together and used to interpret outputs with logic ‘1’ combination. In SOP form each product term is identified as a minterm.

**Example: – ABC’+A’BC+ABC**

110 011 111

** Product of Sum (POS) Form**

In POS or Product of Sum, form variables are ORed(i.e added) together and applied to represent outputs with logic ‘0’ combination. In POS form every single term is known as the maxterm.

**Example: –** (A+B’+C) (A’+B+C’) (A+B+C)

010 101 000

**Gray Code**

Gray Code is defined as the code in which only one bit in the code group changes at a time when proceeding from one step to the other step.

Know more about the ROM.

**Type of K-Map**

**Two-Variable K-Map**

The number of cells in a two-variable K-map is four as the number of variables is two, i.e. a two-variable K-map will have either four minterms or four maxterms. The below figure shows two variable K-Map with SOP and POS representation.

**Three-Variable K-map **

The three-variable K-map is represented by an array of eight cells(eight minterms or eight maxterms). For understanding purposes, we will use A, B, and C as the three variables. However one can use any letter for the names of the variables. The below figure presents three variables K-Map with SOP and POS representation.

If you are reading Karnaugh Maps, you should also read about the Ex-OR GATE.

**Four-Variable K-map**

The four-variable K-map is expressed as an array of 16 cells i.e. sixteen minterms or sixteen maxterms. The below figure presents a four-variable K-Map with SOP and POS representation.

**Five-Variable K-map**

It involves 32 cells; this means that there are 32 minterms or 32 maxterms in a five variable K-map. The below figure presents a five-variable K-Map with SOP representation.

**How to Solve K-map?**

- Choose K-map according to the number of variables.
- Then classify minterms or maxterms as provided in the problem.
- For SOP, place 1’s in K-map cells corresponding to the minterms (0’s elsewhere).
- For POS, put 0’s in cells of K-map corresponding to the maxterms(1’s elsewhere).
- Construct rectangular groups containing total terms in the power of two like 2,4,8 ..(except 1) and attempt to cover as many elements as possible in one group.
- From the groups created in the above steps find the product terms and sum them up for the SOP form and find the sum of terms and product them in POS form.
- The simplification of logical functions applying the K-map is based on the principle of coupling terms in adjacent cells.

Also, learn about AND Gate here.

**Looping**

The method of combining terms in neighboring cells is called looping. In looping groups are created in a combination of 2, 4,8,16, and so on. By folding the k-map over its edges the number of 1’s and 0’s get overlapped, developing a group.

Looping groups of two components are called **pairs**. Looping groups of four components are called **Quads, **whereas looping groups of eight components are called **Octets**.

**Karnaugh Map Solved Example**

- Build the K-map by putting the 1’s in those cells resembling the 1’s in the truth table and place 0’s elsewhere (here standard SOP form is taken).
- After placing the 1’s and 0’s, look for the adjacent 1’s that can be joined for group development. To start with look for octets then quad and lastly pairs.
- The loops and pairs can include 1’s that have previously been paired, make sure to form the least number of loops.
- Lastly, construct the OR sum of all the terms formed by each loop. Consider an example below. In this K-map there is one group of eight(octets) and two groups of four(quad)-one formed. The minimized SOP form is as follows.

F= C’+ A’D’ + BD’

Octal quad 1 quad 2

**Don’t Care Condition**

Some logic circuits can be constructed in such a way that there are a certain number of input combinations for which the output is not fixed, i.e the output can take any value either ‘0’ or ‘1’ depending upon the developer for making the simplification easy. Such conditions are termed don’t care conditions.

Know more about the NAND Gate here.

**Implicants, Prime Implicants, and Essential Prime Implicants.**

- Implicants can be understood as are all the individual elements of each cell in the K-map whereas Prime Implicants is the smallest possible product term of the given function where eliminating any one element is not possible.
- Essential prime Implicant is a prime implicant that should hold at least one minterm that is not included by any other prime implicant.
- Consider an example for better understanding. There are five implicants, three prime implicants, and two essential prime implicants in this example.

- Implicant- A’B’C’, A’B’C, A’BC’, ABC’, ABC.
- Prime Implicant: – A’B’, AB, BC’.
- Essential prime implicant: – A’B’, AB

**Points to Remember **

- K-map gives minimized expression but not sure a unique one.
- In K-map gray code representation is practiced.
- Two K-maps are said to be equal if 1’s are placed in the identical position on both the maps, furnishing equal logical expression.
- Two K-maps are said to be a compliment if one K-map has 1’s and another K-map has 0’s on the same spot.

We hope that the above article on the Karnaugh Maps is helpful for your understanding and exam preparations. Stay tuned to the Testbook app for more updates on related topics from Digital Electronics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams.

**Karnaugh Maps FAQs**

**Q.1 What is the K-map method?**

**Ans.1**The K-map is an approach to solving the logical expressions which are referred to as the graphical method of simplifying Boolean expressions. K-maps deal with the method of inserting the values of the output variable in cells inside a rectangle or square grid according to a specific pattern.

**Q.2 How many types of looping can be done in 3 variables K-maps?**

**Ans.2**Three types of looping can be done in a K-map; Pairs(group of two), Quad(group of four), Octets(group of eight).

**Q.3 What are SOP and POS?**

**Ans.3**The SOP (Sum of Product) and POS (Product of Sum) are the techniques for concluding a particular logic function. In SOP form, products of the element are added together and in POS form, the addition of terms are ANDed together.

**Q.4 Why is Gray code used in the K map?**

**Ans.4**The property of gray code is used in K-map to simplify Boolean functions. The cells in a K-map are numbered in Gray code so that only one bit is modified at a time and one can take variables of consecutive cells in common to drop out the varying bit.

**Q.5 What are Minterm and maxterm?**

**Ans.5**In Minterm, we look for the functions where the output results in “1” while in Maxterm we look for functions where the output results in “0”. We perform Sum of minterm also known as Sum of products (SOP). We perform the Product of Maxterm, also known as the Product of sum (POS).

**Q.6 Is a Karnaugh map a truth table?**

**Ans.6**Karnaugh Map or K-Map is one approach to come up with a minimized SOP or POS formula for a boolean function. It progresses by building a truth table with groupings of variables, which serves to minimize the results when formulating a sum of products or products of sum form.