While studying digital techniques, it is important to understand boolean algebra along with its theorems. These theorems can help you simplify the most-complex boolean expressions and find the logical solution. In this article, we will talk about the Duality Theorem or the Duality Principle in Boolean Algebra.
Algebra is a symbolic form of a mathematical statement. Boolean Algebra is one that is used in digital electronics. While studying electronics, you must try to simplify the logic as far as possible to make implementation easier. When you have a boolean expression, you can simplify it by using one of the several theorems and laws in Boolean Algebra. One such theorem is the Duality Theorem.
Principle of Duality Theorem
Duality Theorem states that “The Dual of the expression can be achieved by replacing the AND operator with OR operator, along with replacing the binary variables, such as replacing 1 with 0 and replacing 0 with 1.“
In other words, the value of the Boolean function doesn’t change even if variables are replaced. However, changing the names of variables also requires you to change binary operators. This brings us to the definition of Duals:
“Duals are the operators and variables of an equation which though interchanged produce no change in the output.“
The Duality Theorem is also called De Morgan Duality which states that:
“Interchanging of Duals pairs in Boolean algebra will result in the same output of the equation.“
Here is a list of operators and variables along with their duals:
|Variable / Operator||Dual|
Further, there is a special type of duality operation – The Self Dual. This operation processes the input to the output without any changes. Sometimes it is called the ‘Do Nothing Operation’.
Solved Examples of Duality Theorem
We start with a simple example.
Find the dual of the statement: 1 + 0 = 1
To find the dual, we follow these steps:
- change the first 1 to 0
- replace the OR (+) operator with the AND (*) operator
- change the first 0 to 1
- change the second 1 to 0
Hence, the dual is: 0 * 1 = 0
Let’s say that we have a Boolean equation: A + B = 0. Applying the Duality Theorem and replacing the variable 1 with 0 and the OR operator with the AND operator, we get A * B = 1. Hence, both the Boolean functions represent the operation of the logical circuit.
We can also say that if A and B are two variables, then A + B = 0 and A * B = 1 are true for the same logic circuit.
You can also use the Duality principle to simplify a Boolean expression. Here is an example of the same:
(A +B’ C)’ = A’ BC + A’ BC’ + A’ B’ C’ = A’ B (C + C’) + (B + B’) A’ C’ = A’ B + A’ C’ – – – – – – – -> (1)
Let’s take the inverse of both sides. We have,
(A +B’ C) = (A + B’) (A + C) – – – – – – – -> (2)
A quick look at equations (1) and (2) shows that the OR and operators are interchanged. This proves the Duality Theorem. It is important to note that you can simplify Boolean functions using two methods of the Duality principle:
- Maximum terms – SOP or Sum Of Products. In this method, you write the maximum terms of the Boolean variables as their sum of products.
- Minimum terms – POS or Product of Sums. In this method, you write the minimum terms of the Boolean variables as their product of sums.