Principle of Duality
Principle of Duality :
Principle of Duality is based on the Boolean algebra and concepts of boolean algebra.
In the boolean algebra, we can choose any symbol according to our convenience. We do not have to depend upon the symbols which are already been given to us for practice. Here it is not necessary to take the symbols as 0 or 1, we can take or use any symbols instead of them like ἀ and ᵦ etc. And as long as we do it continuously it will still be called as Boolean algebra.
We can also convert or use 0 and 1 as 1 and 0 and it would still be called as boolean algebra but with some differences like after this ^ will become ᵥ and now all the operations will give us opposite results.
By this we conclude that we can change the names of the values but if we even change the operations performed with the values then the result will be totally unpredictable that means we cannot assume what will be the output of the values given.
When values and operations are paired in a way that the result or the output or everything important unchanged then we call these pairs as dual to each other. thus with this we can easily understand that 0 an 1 are dual as well as ᵥ and ^ are also dual.
The dual principle or principle of duality says that the boolean algebra remains unchanged when the dual pairs are interchanged.
But nothing goes with compliment because compliment is as self dual operation.
The principle of duality can be explained more effectively with the help of a group theory which states that there are exactly four functions which are one to one mappings from the set of boolean algebra. These four functions are :
- Compliment function
- identity function
- dual function
- contradual function (complimented dual)
These four functions form a group under the set of boolean algebra.
Principle of Duality is a very important principle for Boolean algebra. Generally we just use the principle for solving any boolean algebra, but we show very less interest in knowing about the principle in details. This article is very useful as it gives the details about this principle.