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

  • (For e.g. vector space, incidence configuration, linear program as well as a 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 boolean algebra is.

  • 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 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

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 that 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.


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FAQs (Frequently Asked Questions)

1. What is a duality with example? What is meant by duality in boolean algebra?

In mathematics, the concept of duality can be defined as the principle whereby one true statement can be obtained from another statement by merely interchanging two words. You can see the principle of duality in projective geometry, set theory, as well as symbolic logic are examples of systems with underlying lattice structures.

 

Duality Theorem in Boolean Theorem - Duality theorem states that the dual of the Boolean function can be easily obtained by interchanging the logical AND operator with the logical OR operator and the zeros with ones and vice versa.

2. What is the duality of a 11?

1 equals 1: it is a true statement asserting that "true and true results to true". (d) 0 + 0 equals 0 : (d) is the dual of (c): it is a true statement asserting, correctly, that "false or false will always evaluate to false".

3. What is duality in mathematics?

A duality in mathematics is a one-to-one translation of concepts, theorems, or mathematical structures into other concepts, theorems, or structures, frequently (but not always) using an involution operation: if B is the dual of A, then the dual of B is A. Such involutions can have fixed points, resulting in A's dual being A. Desargues' theorem, for example, is self-dual in this sense under projective geometry's standard duality.


Duality has a variety of interpretations in mathematics. It has been regarded as a pervasive and important notion in mathematics, as well as a major general theme with applications in practically every branch of the subject.


Many mathematical dualities between objects of two kinds relate to pairings, bilinear functions from one type to another type to a family of scalars. For example, linear algebra duality corresponds to bilinear maps from pairs of vector spaces to scalars, distributions, and associated test functions duality corresponds to the pairing of integrating a distribution against a test function, and Poincaré duality corresponds to intersection number, which is viewed as a pairing between submanifolds of a given manifold. For more information, click here.

4. How is duality used in projective geometry?

It is feasible to identify geometric transformations in some projective planes that map each point of the projective plane to a line, and each line of the projective plane to a point, while keeping the incidence. A general principle of duality in projective planes emerges for such planes: given any theorem in such a plane projective geometry, replacing the terms "point" and "line" everywhere results in a new, equally valid theorem. The statement "two points determine a unique line, the line passing through these points," for example, has the dual statement "two lines determine a unique point, the intersection point of these two lines."


The dual vector space provides a conceptual explanation for this phenomenon in some planes (most notably field planes). In fact, the projective plane's points correspond to one-dimensional subvector spaces, whereas the plane's lines belong to two-dimensional subvector spaces W. The duality in such projective geometries arises from giving to a one-dimensional V the subspace of linear maps that fulfill. This space is two-dimensional as a result of the linear algebra dimension formula, i.e. it becomes corresponding to a line in the projective plane.

5. What is duality in algebraic and arithmetic geometry?

Using l-adic cohomology with Ql-coefficients instead, the same duality pattern applies for a smooth projective variety over a separably closed field. This is extended to possibly singular varieties by employing intersection cohomology instead, resulting in a duality known as Verdier duality. Serre duality or coherent duality is identical to the previous claims, but they apply to the cohomology of coherent sheaves.


The new age formulation of these dualities can be seen using derived categories and certain direct and inverse image functors of sheaves (with respect to the classical analytical topology on manifolds for Poincaré duality, l-adic sheaves and the étale topology in the second case, and with respect to coherent sheaves for cohesion duality).

In arithmetic, étale cohomology of finite, local, and global fields (also known as Galois cohomology, because étale cohomology over a field is equivalent to group cohomology over the field's (absolute) Galois group) admits comparable pairings. The profinite completion of Z, the integers, is isomorphic to the absolute Galois group G(Fq) of a finite field. As a result, it's the ideal match.

6. How is duality used in spaces and functions?

A duality between commutative C*-algebras A and compact Hausdorff spaces is known as Gelfand duality. The space of continuous functions (which disappear at infinity) from X to C, the complex numbers, is assigned to X. In contrast, the spectrum of A may be reconstructed from space X. Gelfand and Pontryagin’s duality can both be derived in a large formal, category-theoretic manner.


In algebraic geometry, there is a duality between commutative rings and affine schemes: there is an affine spectrum, Spec A, for every commutative ring A. By taking global sections of the structural sheaf OS, one can receive back a ring given an affine scheme S. Furthermore, ring homomorphisms have a one-to-one relationship with affine scheme morphisms, implying equivalence.

7. What are Galois connections?

In a number of cases, the two categories that are dual to each other are really derived from partially ordered sets, i.e., there is a sense that one object is "smaller" than another. A Galois link is a duality that respects the orderings in question. The typical duality in Galois theory given in the introduction is an example: a larger field extension corresponds to a smaller group under the mapping that assigns the Galois group Gal (/ L) to any extension L K (within any fixed bigger field).


A full Heyting algebra is formed by the collection of all open subsets of a topological space X. There is a duality that connects sober environments and spatial places, known as Stone duality. Head to the Vedantu app and website for gaining access to free study materials.

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