# Duality

## What is Duality?

Duality is known to be a very general as well as a broad concept, without a strict definition that captures all those uses. There usually is a precise definition when duality is applied to specific concepts, for just that context. The common idea is that there are two things that basically are just two sides of the same coin.

Common Themes in this Topic Include:

• We can say one common theme is two different interpretations or descriptions of fundamentally the same structure or object
(For e.g. roles of points as well as lines interchanged, roles of variables in LP changed)

• Primal, as well as dual, often are the same kind of object
(e.g. vector space, incidence configuration, linear program as well as planar graph, etc.)

The general idea of the concept of duality usually is still the same, though every use of the word strictly satisfies all of these given aspects.

### Duality Principle in Boolean Algebra

Let’s first know what is boolean algebra.

• Boolean Algebra is defined as algebra, which deals with binary numbers and binary variables.

• Hence, it is also known as Binary Algebra.

• The other name for boolean algebra is logical Algebra.

• A mathematician, named George Boole was the one who had developed this algebra in the year 1854.

• The variables used in this algebra are known as Boolean variables.

• The boolean variables are 0 and 1.

• In the terms of voltage, the range of voltages corresponding to Logic ‘High’ is represented with the number 1, and the range of voltages corresponding to logic ‘Low’ is represented with the number 0.

## Operator/Variable and Their Duality

 Operator / Variable Dual of the Operator AND OR OR AND 1 1 0 1 A A̅ A̅ A

### Duality Principle

• According to the duality principle, if we have postulates or if we have theorems of Boolean Algebra for any one type of operation then the operation can be converted into another type of operation.

•  In other words  AND can be converted to OR and OR can be converted into AND

• We can interchange '0 with 1', '1 with 0', '(+) sign with (.) sign' and '(.) sign with (+) sign' to perform dual operation. T

• This principle ensures if a theorem is proved using postulates of Boolean algebra, then the dual of this theorem automatically holds and there is no requirement of proving it separately.

The dual of a Boolean expression can easily be obtained by interchanging sums and products and interchanging 0 as well as 1. Let’s know how to find the dual of any expression.

For example, the dual of xy̅ + 1 is equal to (x + y) · 0

Duality Principle: The Duality principle states that when both sides are replaced by their duals the Boolean identity remains valid.

Some Boolean expressions and their corresponding duals are given in the table below:

## Boolean Expressions and Their Corresponding Duals

 Given Expression Dual Given Expression Dual 0 = 1 1 = 0 A. (A+B) = A A + A.B = A 0.1 = 0 1 + 0 = 1 AB = A + B A+B = A.B A.0 = 0 A + 1 = 1 (A+C) (A +B) = AB + AC AC + AB = (A+B). (A+C) A.B = B. A A + B = B + A A+B = AB + AB +AB AB = (A+B).(A+B).(A+B) A.A = 0 A + A = 1 AB + A + AB = 0 ((A+B)).A.(A+B) = 1 A. (B.C) = (A.B). C A+(B+C) = (A+B) + C

### What is Duality in Mathematics?

• In mathematics, we can define duality as a principle that translates concepts, theorems, or mathematical structures into other concepts, theorems, or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of let’s suppose A is equal to B, then we can say that the dual of B is A.

• We can define duality as a property that belongs to the branch of algebra which is known as lattice theory, which is involved with the concepts of order as well as structure common to different mathematical systems.

• Now, what is a lattice? A mathematical structure is known as a lattice if it can be ordered in a specified way.

• Duality in mathematics is basically not a theorem, but we can say it to be a “principle”.

• Duality has a simple origin, the principle is very powerful and useful, and has a long history going back hundreds of years.

• The concept of duality appears in many subjects in mathematics (geometry, algebra, analysis) as well as in physics.

### Duality in Real Life

As hinted at by the word "dual" within it, in simple English we can understand that the word duality refers to having two parts, we can say often with opposite meanings, like the duality of good and evil (opposites). Let’s suppose if there are two sides to a coin, metaphorically speaking, there's a duality in this too. Peace and war, love and hate, up and down, as well as black and white, are dualities.