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Duality Principle in Boolean Algebra with Examples
The "duality principle" is one of the most important topics that can be studied in Boolean algebra, and this article was written and distributed in order to provide a description of it. In addition to that, I will provide an illustration to more concretely illustrate it. Now, let's get started by defining what it is.
What is duality principle?
The duality principle, also known as the principle of duality, is an essential property that is primarily utilized in the process of proving a variety of theorems that can be found in Boolean algebra.
The duality principle states that in a two-valued boolean algebra, the dual of an algebraic expression can be obtained by exchanging all of the OR and AND operators and then replacing 1 with 0 and 0 with 1. In addition, the dual can be obtained by exchanging all of the AND operators for OR operators.
Before moving on to the section on examples, let's get a grasp on the individual steps that make up the duality theorem. In order to help you get a better grasp on the material, I will first walk you through the steps and then give you an example.
Steps used in the Duality Theorem
Here are some of the main steps used to solve the duality theorem in boolean algebra:
- Change each AND operation to an OR operation.
- Change each OR operation to an AND operation.
- Replace 0 with 1.
- Replace 1 with 0.
To put it more simply, we need to make the following adjustments: for every OR, we need to make it an AND, and for every 0 we need to make it a 1, and vice versa.
In order to fully comprehend the subject at hand, it is now time to examine some relevant examples.
Duality Principle Example
Let's take an example to illustrate how to apply all the steps used in the duality principle in a practical way. This example gives you some ideas about how to convert or apply the duality theorem or principle to any boolean expression.
1+0 = 1
The dual of the preceding statement is:
0.1 = 0
As you can see here, we have done the following:
- changed first 1 to 0
- changed OR (+) to AND (.)
- changed first 0 to 1
- changed second 1 to 0
Now, let me demonstrate several examples of the duality principles by using the table that is provided below. In this table, the boolean expression will be written in the first column, and the duals of that expression will be written in the second column.
| Expression | Dual |
|---|---|
| 1 = 0 | 0 = 1 |
| 0 = 1 | 1 = 0 |
| 1.0 = 0 | 0 + 1 = 1 |
| 0.1 = 0 | 1 + 0 = 1 |
| 1 + 0 = 1 | 0.1 = 0 |
| 0 + 1 = 1 | 1.0 = 0 |
| A.0 = 0 | A + 1 = 1 |
| 0.A = 0 | 1 + A = 1 |
| A.1 = 0 | A + 0 = 1 |
| 1.A = 0 | 1 + A = 1 |
| A.A = 0 | A + A = 1 |
| A.B = B.A | A + B = B + A |
| A.(B.C) = (A.B).C | A + (B + C) = (A + B) + C |
| A.(A + B) = A | A + A.B = A |
| AB + C + BCA = 0 | (A + B).C.(B + C + A) = 1 |
Regarding the "duality principle," I believe that we have now covered sufficient ground in terms of examples and specifics. You are welcome to get in touch with us at any time if you have any inquiries, questions, or suggestions regarding this article or any other one. The link to the contact page can be found at the very bottom of the article.
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