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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:
(For e.g. roles of points as well as lines interchanged, roles of variables in LP changed)
(For e.g. vector space, incidence configuration, linear program as well as a planar graph, etc.)
Duality Principle in Boolean Algebra
Let’s first know what boolean algebra is.
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In 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
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We can interchange ‘0 with 1’, ‘1 with 0’, ‘(+) sign with (.) sign’ and ‘(.) sign with (+) sign’ to perform dual operation. T
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?
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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.
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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.
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Duality has a simple origin, the principle is very powerful and useful, and has a long history going back hundreds of years.
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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.