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How to Simplify Logic Functions Using Karnaugh Maps

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  • 0:04 Karnaugh Maps
  • 0:46 2 Variable Truth Tables
  • 1:40 3 Variable Truth Tables
  • 2:02 4 Variable Truth Tables
  • 2:40 Higher Order K-Maps
  • 3:12 Lesson Summary
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Lesson Transcript
Instructor: Jim Stiles

Jim Stiles holds a BS in Electrical Engineering & is a licensed Professional Engineer. Jim has worked in electronic circuit design, test & SW development for over 20 years.

In this lesson we are going to learn how to use Karnaugh Maps to simplify Boolean logic. The resulting Boolean equation represents a minimized function suitable for implementation.

Karnaugh Maps

You've come up with a brilliant idea that solves a problem you have been working on for weeks. You have identified a detailed series of logical operations that will result in the perfect solution. The only problem is that it is pretty complicated, and you need a clever way to simplify things. What you need is a Karnaugh Map.

Karnaugh or K-Maps are used to simplify and minimize the number of logical operations required to implement a Boolean function. The implementation of the function may be in the form of digital electronic hardware or software statements.

A Boolean function is an algebraic expression with variables that represent the binary values 0 and 1. Some useful Boolean identities and laws are listed below.


Boolean Equations & Useful Identities & Laws


2 Variable Truth Table and K-Map

A logical specification is often created using a truth table. A truth table is a list of the inputs (A, B) on the left and the corresponding output (F) on the right. See Figure 1 showing a 2 variable truth table and corresponding K-Map.


Figure 1: Variable Truth Table & K-Map
2 Variable Truth Table & K-Map


A K-Map creates a new representation of the truth table using Gray code ordering, ensuring that only one bit changes for any adjacent cell. It is a logical adjacency that makes Boolean simplification possible.

Each cell of the K-Map represents an input state (A, B). The value of each cell represents the output function (F). In order to find the minimum logic function, it is necessary to identify matching adjacent cells. Once these matches are found, an expression can be written. See Figure 2 showing how to group and determine the function F for the K-Map outlined in Figure 1.


Figure 2: 2 Variable K-Map Loop & Find Function F
2 Variable K-Map Loop & Find Function F


Note : This lesson will be using the sum of products (SOP) form for expressions. This is achieved by minimizing the logical 1s in the K-Maps. It is also possible to use the product of sums (POS) form by minimizing the logical 0s.

3 Variable Truth Table and K-Map

Below is an example of a 3 variable K-Map. Notice that the cells are ordered in the K-Map to ensure only one bit changes on any adjacent cell. From left to right instead of 0, 1, 2, 3, 4, 5 ,6, 7, the cell ordering is 0, 1, 2, 3, 6, 7, 4, 5.

000 010 110 100
001 011 111 101


Figure 3: Variable Truth Table & K-Map
3 Varible Truth Table & K-Map


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