Mathematical Relations IP

Mathematical relations IPFrom Wikipedia, the free encyclopedia

Contents

1 Inverse relation 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Inverse relation of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Inverse trigonometric functions 32.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Etymology of the arc- prex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 Relationships between trigonometric functions and inverse trigonometric functions . . . . . 42.2.3 Relationships among the inverse trigonometric functions . . . . . . . . . . . . . . . . . . . 42.2.4 Arctangent addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 In calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3.1 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 52.3.2 Expression as denite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.3 Innite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3.4 Indenite integrals of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . 7

2.4 Extension to complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4.1 Logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5.2 In computer science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Near sets 173.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Nearness of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Generalization of set intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

i

ii CONTENTS

3.4 Efremovi proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Visualization of EF-axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Descriptive proximity space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.7 Proximal relator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.8 Descriptive -neighbourhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.9 Tolerance near sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.10 Tolerance classes and preclasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.10.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.11 Nearness measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.12 Near set evaluation and recognition (NEAR) system . . . . . . . . . . . . . . . . . . . . . . . . . 293.13 Proximity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.14 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.17 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Partial equivalence relation 364.1 Properties and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.1 Euclidean parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Kernels of partial functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2.3 Functions respecting equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Partial function 385.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Total function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3 Discussion and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.1 Natural logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.2 Subtraction of natural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.3 Bottom element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.4 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.3.5 In abstract algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Partially ordered set 416.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.3 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426.4 Orders on the Cartesian product of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . 43

CONTENTS iii

6.5 Sums of partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.6 Strict and non-strict partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.7 Inverse and order dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.8 Mappings between partially ordered sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.9 Number of partial orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.10 Linear extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.11 In category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.12 Partial orders in topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.13 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.14 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.15 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.17 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

7 Preorder 487.1 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.4 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.5 Number of preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507.6 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

8 Prewellordering 528.1 Prewellordering property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528.1.2 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

8.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

9 Propositional function 549.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 55

9.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Chapter 1

Inverse relation

For inverse relationships in statistics, see negative relationship.

In mathematics, the inverse relation of a binary relation is the relation that occurs when the order of the elements isswitched in the relation. For example, the inverse of the relation 'child of' is the relation 'parent of'. In formal terms,if X and Y are sets and L X Y is a relation from X to Y then L1 is the relation dened so that y L1 x if andonly if xLy . In set-builder notation, L1 = f(y; x) 2 Y X j (x; y) 2 Lg .The notation comes by analogy with that for an inverse function. Although many functions do not have an inverse;every relation does have a unique inverse. Despite the notation and terminology, the inverse relation is not an inversein the sense of group inverse; the unary operation that maps a relation to the inverse relation is however an involution,so it induces the structure of a semigroup with involution on the binary relations on a set, or more generally induces adagger category on the category of relations as detailed below. As a unary operation, taking the inverse (sometimescalled inversion) commutes however with the order-related operations of relation algebra, i.e. it commutes withunion, intersection, complement etc.The inverse relation is also called the converse relation or transpose relation the latter in view of its similaritywith the transpose of a matrix.[1] It has also been called the opposite or dual of the original relation.[2] Other notationsfor the inverse relation include LC , LT , L~ or L or L or L.

1.1 ExamplesFor usual (maybe strict or partial) order relations, the converse is the naively expected opposite order, e.g. 1=; , etc.

1.1.1 Inverse relation of a function

A function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inversefunction.The inverse relation of a function f : X ! Y is the relation f1 : Y ! X dened by graph f1 = f(y; x) j y =f(x)g .This is not necessarily a function: One necessary condition is that f be injective, since else f1 is multi-valued. Thiscondition is sucient for f1 being a partial function, and it is clear that f1 then is a (total) function if and only iff is surjective. In that case, i.e. if f is bijective, f1 may be called the inverse function of f.

1.2 PropertiesIn the monoid of binary endorelations on a set (with the binary operation on relations being the composition ofrelations), the inverse relation does not satisfy the denition of an inverse from group theory, i.e. if L is an arbitrary

1

2 CHAPTER 1. INVERSE RELATION

relation on X, then L L1 does not equal the identity relation on X in general. The inverse relation does satisfy the(weaker) axioms of a semigroup with involution: (L1)1 = L and (L R)1 = R1 L1 .[3]Since one may generally consider relations between dierent sets (which form a category rather than a monoid,namely the category of relations Rel), in this context the inverse relation conforms to the axioms of a dagger category(aka category with involution).[3] A relation equal to its inverse is a symmetric relation; in the language of daggercategories, it is self-adjoint.Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relationsas sets), and actually an involutive quantale. Similarly, the category of heterogenous relations, Rel is also an orderedcategory.[3]

In relation algebra (which is an abstraction of the properties of the algebra of endorelations on a set), inversion (theoperation of taking the inverse relation) commutes with other binary operations of union and intersection. Inversionalso commutes with unary operation of complementation as well as with taking suprema and inma. Inversion is alsocompatible with the ordering of relations by inclusion.[1]

If a relation is reexive, irreexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partialorder, total order, strict weak order, total preorder (weak order), or an equivalence relation, its inverse is too.

1.3 See also Bijection Function (mathematics) Inverse function Relation (mathematics) Transpose graph

1.4 References[1] Gunther Schmidt; Thomas Strhlein (1993). Relations and Graphs: Discrete Mathematics for Computer Scientists. Springer

Berlin Heidelberg. pp. 910. ISBN 978-3-642-77970-1.

[2] Celestina Cotti Ferrero; Giovanni Ferrero (2002). Nearrings: Some Developments Linked to Semigroups and Groups.Kluwer Academic Publishers. p. 3. ISBN 978-1-4613-0267-4.

[3] Joachim Lambek (2001). Relations Old and New. In Ewa Orlowska, Andrzej Szalas. Relational Methods for ComputerScience Applications. Springer Science & Business Media. pp. 135146. ISBN 978-3-7908-1365-4.

Halmos, Paul R. (1974), Naive Set Theory, p. 40, ISBN 978-0-387-90092-6

Chapter 2

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions[1]) are the inversefunctions of the trigonometric functions (with suitably restricted domains). Specically, they are the inverses of thesine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of theangles trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, andgeometry.

2.1 NotationThere are many notations used for the inverse trigonometric functions. The notations sin1 (x), cos1 (x), tan1(x), etc. are often used, but this convention logically conicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion betweenmultiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each ofthe reciprocal trigonometric functions has its own namefor example, cos(x)1=sec(x). Another convention usedby some authors[2] is to use a majuscule (capital/upper-case) rst letter along with a 1 superscript, e.g., Sin1 (x),Cos1 (x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin1 (x),cos1 (x), etc. Yet another convention is to use an arc- prex, so that the confusion with the 1 superscript is resolvedcompletely, e.g., arcsin (x), arccos (x), etc. This convention is used throughout the article. In computer programminglanguages (also MS Oce Excel) the inverse trigonometric functions are usually called asin, acos, atan.According to Cajori,[3] the notation sin1 (x) was introduced by John Herschel in 1813.[4]

2.1.1 Etymology of the arc- prex

When measuring in radians, an angle of radians will correspond to an arc whose length is r, where r is the radiusof the circle. Thus, in the unit circle, the arc whose cosine is x is the same as the angle whose cosine is x, becausethe length of the arc of the circle in radii is the same as the measurement of the angle in radians.[5]

2.2 Basic properties

2.2.1 Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions.Therefore the ranges of the inverse functions are proper subsets of the domains of the original functionsFor example, using function in the sense of multivalued functions, just as the square root function y = x could bedened from y2 = x, the function y = arcsin(x) is dened so that sin(y) = x. There are multiple numbers y such thatsin(y) = x; for example, sin(0) = 0, but also sin() = 0, sin(2) = 0, etc. When only one value is desired, the functionmay be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x)

3

4 CHAPTER 2. INVERSE TRIGONOMETRIC FUNCTIONS

will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometricfunctions.The principal inverses are listed in the following table.(Note: Some authors dene the range of arcsecant to be ( 0 y < /2 or y < 3/2 ), because the tangentfunction is nonnegative on this domain. This makes some computations more consistent. For example using thisrange, tan(arcsec(x))=x21, whereas with the range ( 0 y < /2 or /2 < y ), we would have to writetan(arcsec(x))=x21, since tangent is nonnegative on 0 y < /2 but nonpositive on /2 < y . For a similarreason, the same authors dene the range of arccosecant to be ( - < y -/2 or 0 < y /2 ).)If x is allowed to be a complex number, then the range of y applies only to its real part.

2.2.2 Relationships between trigonometric functions and inverse trigonometric functionsTrigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is byconsidering the geometry of a right-angled triangle, with one side of length 1, and another side of length x (any realnumber between 0 and 1), then applying the Pythagorean theorem and denitions of the trigonometric ratios. Purelyalgebraic derivations are longer.

2.2.3 Relationships among the inverse trigonometric functionsComplementary angles:

arccosx = 2 arcsinx

arccotx = 2 arctanx

arccscx = 2 arcsecx

Negative arguments:

arcsin(x) = arcsinxarccos(x) = arccosxarctan(x) = arctanxarccot(x) = arccotxarcsec(x) = arcsecxarccsc(x) = arccscxReciprocal arguments:

arccos 1x = arcsecxarcsin 1x = arccscxarctan 1x = 12 arctanx = arccotx ; if x > 0arctan 1x = 12 arctanx = + arccotx ; if x < 0arccot 1x = 12 arccotx = arctanx ; if x > 0arccot 1x = 32 arccotx = + arctanx ; if x < 0arcsec 1x = arccosxarccsc 1x = arcsinxIf you only have a fragment of a sine table:

2.3. IN CALCULUS 5

arccosx = arcsinp

1 x2 ; if 0 x 1arctanx = arcsin xp

x2 + 1

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positiveimaginary part if the square was negative real).From the half-angle formula tan 2 = sin 1+cos , we get:

arcsinx = 2 arctan x1 +

p1 x2

arccosx = 2 arctanp1 x21 + x

; if 1 < x +1

arctanx = 2 arctan x1 +

p1 + x2

2.2.4 Arctangent addition formula

arctanu+ arctan v = arctanu+ v

1 uv

(mod ) ; uv 6= 1 :

This is derived from the tangent addition formula

tan(+ ) = tan+ tan1 tan tan ;

by letting

= arctanu ; = arctan v :

2.3 In calculus

2.3.1 Derivatives of inverse trigonometric functionsMain article: Dierentiation of trigonometric functions

The derivatives for complex values of z are as follows:

ddz arcsin z =

1p1 z2 ; z 6= 1;+1

ddz arccos z =

1p1 z2 ; z 6= 1;+1

ddz arctan z =

1

1 + z2; z 6= i;+i

ddz arccot z =

11 + z2

; z 6= i;+id

dz arcsec z =1

z2p1 z2 ; z 6= 1; 0;+1

ddz arccsc z =

1z2p1 z2 ; z 6= 1; 0;+1


Only for real values of x:

ddx arcsecx =

1

jxjpx2 1 ; jxj > 1d

dx arccscx =1

jxjpx2 1 ; jxj > 1

For a sample derivation: if = arcsinx , we get:

d arcsinxdx =

dd sin =

dcos d =

1

cos =1p

1 sin2 =

1p1 x2

2.3.2 Expression as denite integrals

Integrating the derivative and xing the value at one point gives an expression for the inverse trigonometric functionas a denite integral:

arcsinx =Z x0

1p1 z2 dz ; jxj 1

arccosx =Z 1x

1p1 z2 dz ; jxj 1

arctanx =Z x0

1

z2 + 1dz ;

arccotx =Z 1x

1

z2 + 1dz ;

arcsecx =Z x1

1

zpz2 1 dz ; x 1

arcsecx = +Z 1x

1

zpz2 1 dz ; x 1

arccscx =Z 1x

1

zpz2 1 dz ; x 1

arccscx =Z x1

1

zpz2 1 dz ; x 1

When x equals 1, the integrals with limited domains are improper integrals, but still well-dened.

2.3.3 Innite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using innite series, asfollows:

arcsin z = z +1

2

z3

3+

1 32 4

z5

5+

1 3 52 4 6

z7

7+ =

1Xn=0

2nn

z2n+1

4n(2n+ 1); jzj 1

arccos z = 2 arcsin z =

2z +

1

2

z3

3+

1 32 4

z5

5+

=

2

1Xn=0

2nn

z2n+1

4n(2n+ 1); jzj 1

arctan z = z z3

3+z5

5 z

7

7+ =

1Xn=0

(1)nz2n+12n+ 1

; jzj 1 z 6= i;i

2.3. IN CALCULUS 7

arccot z = 2arctan z =

2z z

3

3+z5

5 z

7

7+

=

2

1Xn=0

(1)nz2n+12n+ 1

; jzj 1 z 6= i;i

arcsec z = arccos(1/z) = 2z1 +

1

2

z3

3+

1 32 4

z5

5+

=

21Xn=0

2nn

z(2n+1)

4n(2n+ 1); jzj 1

arccsc z = arcsin(1/z) = z1 +1

2

z3

3+

1 32 4

z5

5+ =

1Xn=0

2nn

z(2n+1)

4n(2n+ 1); jzj 1

Leonhard Euler found a more ecient series for the arctangent, which is:

arctan z = z1 + z2

1Xn=0

nYk=1

2kz2

(2k + 1)(1 + z2):

(Notice that the term in the sum for n = 0 is the empty product which is 1.)Alternatively, this can be expressed:

arctan z =1Xn=0

22n(n!)2

(2n+ 1)!

z2n+1

(1 + z2)n+1

Variant: Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

arctan z = z

1 +(1z)2

3 1z2 + (3z)2

5 3z2 + (5z)2

7 5z2 + (7z)2

9 7z2 + . . .

=z

1 +(1z)2

3 +(2z)2

5 +(3z)2

7 +(4z)2

9 +. . .

The second of these is valid in the cut complex plane. There are two cuts, from i to the point at innity, going downthe imaginary axis, and from i to the point at innity, going up the same axis. It works best for real numbers runningfrom 1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the rst) arejust (nz)2, with each perfect square appearing once. The rst was developed by Leonhard Euler; the second by CarlFriedrich Gauss utilizing the Gaussian hypergeometric series.

2.3.4 Indenite integrals of inverse trigonometric functions

For real and complex values of x:


Zarcsinx dx = x arcsinx+

p1 x2 + CZ

arccosx dx = x arccosxp

1 x2 + CZarctanx dx = x arctanx 1

2ln1 + x2

+ CZ

arccotx dx = x arccotx+ 12

ln1 + x2

+ CZ

arcsecx dx = x arcsecx ln"x

1 +

rx2 1x2

!#+ C

Zarccscx dx = x arccscx+ ln

"x

1 +

rx2 1x2

!#+ C

For real x 1:

Zarcsecx dx = x arcsecx ln

x+

px2 1

+ CZ

arccscx dx = x arccscx+ lnx+

px2 1

+ C

All of these can be derived using integration by parts and the simple derivative forms shown above.

Example

UsingRu dv = uv R v du , set

u = arcsinx dv = dx

du = dxp1 x2 v = x

Then

Zarcsin(x) dx = x arcsinx

Zxp

1 x2 dx

Substitute

k = 1 x2 :Then

dk = 2x dxand

Zxp

1 x2 dx = 1

2

Z dkpk

= pk

Back-substitute for x to yield

Zarcsin(x) dx = x arcsinx+

p1 x2 + C

2.4. EXTENSION TO COMPLEX PLANE 9

2.4 Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complexplane. This results in functions with multiple sheets and branch points. One possible way of dening the extensionsis:

arctan z =Z z0

dx1 + x2

z 6= i;+i

where the part of the imaginary axis which does not lie strictly between i and +i is the cut between the principalsheet and other sheets;

arcsin z = arctan zp1 z2 z 6= 1;+1

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does notlie strictly between 1 and +1 is the cut between the principal sheet of arcsin and other sheets;

arccos z = 2 arcsin z z 6= 1;+1

which has the same cut as arcsin;

arccot z = 2 arctan z z 6= i;+i

which has the same cut as arctan;

arcsec z = arccos 1z

z 6= 1; 0;+1

where the part of the real axis between 1 and +1 inclusive is the cut between the principal sheet of arcsec and othersheets;

arccsc z = arcsin 1z

z 6= 1; 0;+1

which has the same cut as arcsec.

2.4.1 Logarithmic forms

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain tothe complex plane.


arcsinx = i ln

ix+p

1 x2

= arccsc 1x

arccosx = i lnx i

p1 x2

=

2+ i ln

ix+

p1 x2

=

2 arcsinx = arcsec 1

x

arctanx = 12 i (ln (1 ix) ln (1 + ix)) = arccot1

x

arccotx = 12 i

ln1 i

x

ln

1 +

ix

= arctan 1

x

arcsecx = i ln

ir

1 1x2

+1

x

!= i ln

r1 1

x2+

ix

!+

2=

2 arccscx = arccos 1

x

arccscx = i ln r

1 1x2

+ix

!= arcsin 1

x

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof

= arcsinxsin() = sin(arcsinx)sin() = xUsing the exponential denition of sine

ei ei2i = sin()

one obtains

ei ei2i = x

Let

k = ei

Then

k 1k2i = x

k 1k

= 2ix

k 2ix 1k

= 0

k2 2 i k x 1 = 0k = ix

p1 x2

2.5. APPLICATIONS 11

ei = ixp

1 x2

i = ln

ixp

1 x2

= i ln

ixp

1 x2

(the positive branch is chosen)

= arcsinx = i ln

ix+p

1 x2

Example proof (variant 2)

= arcsinxei = cos() + i sin()Apply the natural logarithm, multiply by -i and substitute theta.arcsinx = i ln(cos(arcsinx) + i sin(arcsinx))arcsinx = i ln(p1 x2 + ix)

2.5 Applications

2.5.1 General solutionsEach of the trigonometric functions is periodic in the real part of its argument, running through all its values twice ineach interval of 2. Sine and cosecant begin their period at 2k /2 (where k is an integer), nish it at 2k + /2,and then reverse themselves over 2k + /2 to 2k + 3/2. Cosine and secant begin their period at 2k, nish it at2k + , and then reverse themselves over 2k + to 2k + 2. Tangent begins its period at 2k /2, nishes it at2k + /2, and then repeats it (forward) over 2k + /2 to 2k + 3/2. Cotangent begins its period at 2k, nishesit at 2k + , and then repeats it (forward) over 2k + to 2k + 2.This periodicity is reected in the general inverses where k is some integer:

sin(y) = x , y = arcsin(x) + 2k or y = arcsin(x) + 2k

sin(y) = x , y = (1)k arcsin(x) + kcos(y) = x , y = arccos(x) + 2k or y = 2 arccos(x) + 2kcos(y) = x , y = arccos(x) + 2ktan(y) = x , y = arctan(x) + kcot(y) = x , y = arccot(x) + ksec(y) = x , y = arcsec(x) + 2k or y = 2 arcsec(x) + 2kcsc(y) = x , y = arccsc(x) + 2k or y = arccsc(x) + 2k

Application: nding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle whenthe lengths of the sides of the triangle are known. Recalling the right-triangle denitions of sine, for example, itfollows that

= arcsin opposite

hypotenuse

:


Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using thePythagorean Theorem: a2 + b2 = h2 where h is the length of the hypotenuse. Arctangent comes in handy inthis situation, as the length of the hypotenuse is not needed.

= arctanopposite

adjacent

:

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle with the horizontal, where may be computed as follows:

= arctanopposite

adjacent

= arctan

riserun

= arctan

8

20

21:8 :

2.5.2 In computer science and engineeringTwo-argument variant of arctangent

Main article: atan2

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (, ]. Inother words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positivesign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane,y < 0). It was rst introduced in many computer programming languages, but it is now also common in other eldsof science and engineering.In terms of the standard arctan function, that is with range of (/2, /2), it can be expressed as follows:

atan2(y; x) =

8>>>>>>>>>>>>>>>:

arctan( yx ) x > 0arctan( yx ) + y 0 ; x < 0arctan( yx ) y < 0 ; x < 02 y > 0 ; x = 0

2 y < 0 ; x = 0undened y = 0 ; x = 0

It also equals the principal value of the argument of the complex number x + iy.This function may also be dened using the tangent half-angle formulae as follows:

atan2(y; x) = 2 arctan ypx2 + y2 + x

provided that either x > 0 or y 0. However this fails if given x 0 and y = 0 so the expression is unsuitable forcomputational use.The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such asthe C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted.These variations are detailed at atan2.

Arctangent function with location parameter

In many applications the solution y of the equation x = tan y is to come as close as possible to a given value1 <

2.6. SEE ALSO 13

Practical considerations

For angles near 0 and , arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in acomputer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near /2and /2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

2.6 See also Argument (complex analysis) Complex logarithm Gausss continued fraction Inverse hyperbolic function List of integrals of inverse trigonometric functions List of trigonometric identities Square root Tangent half-angle formula Trigonometric function

2.7 References[1] For example Drrie, Heinrich (1965). Triumph der Mathematik. Trans. David Antin. Dover. p. 69. ISBN 0-486-61348-8.

[2] Prof. Sanaullah Bhatti; Ch. Nawab-ud-Din; Ch. Bashir Ahmed; Dr. S. M. Yousuf; Dr. Allah Bukhsh Taheem (1999).Dierentiation of Tigonometric, Logarithmic and Exponential Functions. In Prof. Mohammad Maqbool Ellahi, Dr.Karamat Hussain Dar, Faheem Hussain. Calculus and Analytic Geometry (in Pakistani English) (First ed.). Lahore: PunjabTextbook Board. p. 140.

[3] Cajori, Florian (1919). A History of Mathematics (2nd ed.). The Macmillan Company, New York. p. 272., at GoogleBooks

[4] Herschel, John F. W. (1813). On a remarkable Application of Cotess Theorem. Philosophical Transactions (RoyalSociety, London) 103 (1): 10., at Google Books

[5] Inverse trigonometric functions in The Americana: a universal reference library, Vol.21, Ed. Frederick Converse Beach,George Edwin Rines, (1912).

2.8 External links Weisstein, Eric W., Inverse Trigonometric Functions, MathWorld. Weisstein, Eric W., Inverse Tangent, MathWorld.


The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.

2.8. EXTERNAL LINKS 15

The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.

Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.


A C

B

b

ah

(adjacent)

(opposite)(hypotenuse)

A right triangle.

Chapter 3

Near sets

Figure 1. Descriptively, very near sets

In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection.In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. De-scriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjointsets. Spatially near sets are also descriptively near sets.The underlying assumption with descriptively close sets is that such sets contain elements that have location andmeasurable features such as colour and frequency of occurrence. The description of the element of a set is denedby a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near

17

18 CHAPTER 3. NEAR SETS

Figure 2. Descriptively, minimally near sets

sets. Near set theory provides a formal basis for the observation, comparison, and classication of elements in setsbased on their closeness, either spatially or descriptively. Near sets oer a framework for solving problems based onhuman perception that arise in areas such as image processing, computer vision as well as engineering and scienceproblems.Near sets have a variety of applications in areas such as topology[37], pattern detection and classication[50], abstract al-gebra[51], mathematics in computer science[38], and solving a variety of problems based on human perception[42][82][47][52][56]that arise in areas such as image analysis[54][14][46][17][18], image processing[40], face recognition[13], ethology[64], aswell as engineering and science problems[55][64][42][19][17][18]. From the beginning, descriptively near sets have provedto be useful in applications of topology[37], and visual pattern recognition [50], spanning a broad spectrum of applica-tions that include camouage detection, micropaleontology, handwriting forgery detection, biomedical image analy-sis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, andtopological psychology.As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colourmodel for varying degrees of nearness between sets of picture elements in pictures (see, e.g.,[17] 4.3). The two pairsof ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the gures corresponds to an equivalenceclass where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals inFig.1 are closer to each other descriptively than the ovals in Fig. 2.

3.1. HISTORY 19

3.1 HistoryIt has been observed that the simple concept of nearness unies various concepts of topological structures[20] inas-much as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetrictopological spaces and continuous maps[3]),Prox (proximity spaces and -maps[8][67]),Unif (uniform spaces and uni-formly continuous maps[81][77]) andCont (contiguity spaces and contiguity maps[24]) as embedded full subcategories[20][59].The categories "ANear and "AMer are shown to be full supercategories of various well-known categories, in-cluding the category sTop of symmetric topological spaces and continuous maps, and the category Met1 ofextended metric spaces and nonexpansive maps. The notation A ,! B reads categoryA is embedded in categoryB. The categories "AMer and "ANear are supercategories for a variety of familiar categories[76] shown in Fig. 3.Let "ANear denote the category of all " -approach nearness spaces and contractions, and let "AMer denote thecategory of all " -approach merotopic spaces and contractions.

Figure 3. Supercats

Among these familiar categories is sTop , the symmetric form of Top (see category of topological spaces), the cat-egory with objects that are topological spaces and morphisms that are continuous maps between them[1][32]. Met1with objects that are extended metric spaces is a subcategory of "AP (having objects " -approach spaces and con-tractions) (see also[57][75]). Let X ; Y be extended pseudometrics on nonempty sets X;Y , respectively. The mapf : (X; X) ! (Y; Y ) is a contraction if and only if f : (X; DX ) ! (Y; DY ) is a contraction. Fornonempty subsets A;B 2 2X , the distance function D : 2X 2X ! [0;1] is dened by

D(A;B) =

(inf f(a; b) : a 2 A; b 2 Bg; ifA and Bempty not are ;1; ifA or Bempty is :

Thus " AP is embedded as a full subcategory in "ANear by the functor F : "AP ! "ANear dened byF ((X; )) = (X; D) and F (f) = f . Then f : (X; X) ! (Y; Y ) is a contraction if and only if f :


(X; DX ) ! (Y; DY ) is a contraction. Thus "AP is embedded as a full subcategory in "ANear by thefunctor F : "AP ! "ANear dened by F ((X; )) = (X; D) and F (f) = f: Since the category Met1of extended metric spaces and nonexpansive maps is a full subcategory of "AP , therefore, "ANear is also a fullsupercategory of Met1 . The category "ANear is a topological construct[76].

Figure 4. Frigyes Riesz, 1880-1956

The notions of near and far[A] in mathematics can be traced back to works by Johann Benedict Listing and FelixHausdor. The related notions of resemblance and similarity can be traced back to J.H. Poincar, who introducedsets of similar sensations (nascent tolerance classes) to represent the results of G.T. Fechners sensation sensitivityexperiments[10] and a framework for the study of resemblance in representative spaces as models of what he termedphysical continua[63][60][61]. The elements of a physical continuum (pc) are sets of sensations. The notion of a pcand various representative spaces (tactile, visual, motor spaces) were introduced by Poincar in an 1894 article onthe mathematical continuum[63], an 1895 article on space and geometry[60] and a compendious 1902 book on science

3.2. NEARNESS OF SETS 21

and hypothesis[61] followed by a number of elaborations, e.g.,[62]. The 1893 and 1895 articles on continua (Pt. 1,ch. II) as well as representative spaces and geometry (Pt. 2, ch IV) are included as chapters in[61]. Later, F. Rieszintroduced the concept of proximity or nearness of pairs of sets at the International Congress of Mathematicians(ICM) in 1908[65].During the 1960s, E.C. Zeeman introduced tolerance spaces in modelling visual perception[83]. A.B. Sossinskyobserved in 1986[71] that the main idea underlying tolerance space theory comes from Poincar, especially[60]. In2002, Z. Pawlak and J. Peters[B] considered an informal approach to the perception of the nearness of physical objectssuch as snowakes that was not limited to spatial nearness. In 2006, a formal approach to the descriptive nearness ofobjects was considered by J. Peters, A. Skowron and J. Stepaniuk[C] in the context of proximity spaces[39][33][35][21].In 2007, descriptively near sets were introduced by J. Peters[D][E] followed by the introduction of tolerance nearsets[41][45]. Recently, the study of descriptively near sets has led to algebraic[22][51], topological and proximity space[37]foundations of such sets.

3.2 Nearness of setsThe adjective near in the context of near sets is used to denote the fact that observed feature value dierences ofdistinct objects are small enough to be considered indistinguishable, i.e., within some tolerance.The exact idea of closeness or 'resemblance' or of 'being within tolerance' is universal enough to appear, quite naturally,in almost any mathematical setting (see, e.g.,[66]). It is especially natural in mathematical applications: practicalproblems, more often than not, deal with approximate input data and only require viable results with a tolerable levelof error[71].The words near and far are used in daily life and it was an incisive suggestion of F. Riesz[65] that these intuitiveconcepts be made rigorous. He introduced the concept of nearness of pairs of sets at the ICM in Rome in 1908. Thisconcept is useful in simplifying teaching calculus and advanced calculus. For example, the passage from an intuitivedenition of continuity of a function at a point to its rigorous epsilon-delta denition is sometime dicult for teachersto explain and for students to understand. Intuitively, continuity can be explained using nearness language, i.e., afunction f : R! R is continuous at a point c , provided points fxg near c go into points ff(x)g near f(c) . UsingRieszs idea, this denition can be made more precise and its contrapositive is the familiar denition[4][36].

3.3 Generalization of set intersectionFrom a spatial point of view, nearness (aka proximity) is considered a generalization of set intersection. For disjointsets, a form of nearness set intersection is dened in terms of a set of objects (extracted from disjoint sets) that havesimilar features within some tolerance (see, e.g., 3 in[80]). For example, the ovals in Fig. 1 are considered near eachother, since these ovals contain pairs of classes that display similar (visually indistinguishable) colours.

3.4 Efremovi proximity spaceLet X denote a metric topological space that is endowed with one or more proximity relations and let 2X denote thecollection of all subsets of X . The collection 2X is called the power set of X .There are many ways to dene Efremovi proximities on topological spaces (discrete proximity, standard proximity,metric proximity, ech proximity, Alexandro proximity, and Freudenthal proximity), For details, see 2, pp. 9394 in[6]. The focus here is on standard proximity on a topological space. For A;B X , A is near B (denoted byA B ), provided their closures share a common point.The closure of a subset A 2 2X (denoted by cl(A) ) is the usual Kuratowski closure of a set[F], introduced in 4, p.20[27], is dened by

cl(A) = fx 2 X : D(x;A) = 0g ; whereD(x;A) = inf fd(x; a) : a 2 Ag :i.e. cl(A) is the set of all points x in X that are close to A ( D(x;A) is the Hausdor distance (see 22, p. 128,


in[15]) between x and the set A and d(x; a) = jx aj (standard distance)). A standard proximity relation is denedby

=(A;B) 2 2X 2X : cl(A) \ cl(B) 6= ; :

Whenever sets A and B have no points in common, the sets are farfrom each other (denoted A B ).The following EF-proximity[G] space axioms are given by Jurij Michailov Smirnov[67] based on what Vadim Arsenye-vi Efremovi introduced during the rst half of the 1930s[8]. Let A;B;E 2 2X .

EF.1 If the set A is close to B , then B is close to A .

EF.2 A [B is close to E , if and only if, at least one of the sets A or B is close to E .EF.3 Two points are close, if and only if, they are the same point.

EF.4 All sets are far from the empty set ; .EF.5 For any two sets A and B which are far from each other, there exists C;D 2 2X , C [D = X , such that A

is far from C and B is far from D (Efremovi-axiom).

The pair (X; ) is called an EF-proximity space. In this context, a space is a set with some added structure. Witha proximity space X , the structure of X is induced by the EF-proximity relation . In a proximity space X , theclosure of A in X coincides with the intersection of all closed sets that contain A .

Theorem 1[67] The closure of any set A in the proximity space X is the set of points x 2 X that are close to A .

3.5 Visualization of EF-axiomLet the set X be represented by the points inside the rectangular region in Fig. 5. Also, let A;B be any two non-intersection subsets (i.e. subsets spatially far from each other) inX , as shown in Fig. 5. LetCc = XnC (complementof the set C ). Then from the EF-axiom, observe the following:

A B;

B C;D = Cc;

X = D [ C;A D; hence, we can write

A B ) A C and B D; for some C;D in X so that C [D = X:

3.6 Descriptive proximity spaceDescriptively near sets were introduced as a means of solving classication and pattern recognition problems arisingfrom disjoint sets that resemble each other[44][43]. Recently, the connections between near sets in EF-spaces and nearsets in descriptive EF-proximity spaces have been explored in[53][48].Again, let X be a metric topological space and let = f1; : : : ; ng a set of probe functions that represent featuresof each x 2 X . The assumption made here is X contains non-abstract points that have measurable features such asgradient orientation. A non-abstract point has a location and features that can be measured (see 3 in [26]).A probe function : X ! R represents a feature of a sample point in X . The mapping : X ! Rn is denedby (x) = (1(x); : : : ; n(x)) , where Rn is an n-dimensional real Euclidean vector space. (x) is a feature vectorfor x , which provides a description of x 2 X . For example, this leads to a proximal view of sets of picture pointsin digital images[48].

3.6. DESCRIPTIVE PROXIMITY SPACE 23

X

AB

CC c

Figure 5. Example of a descriptive EF-proximity relation between sets A;B , and Cc

To obtain a descriptive proximity relation (denoted by ), one rst chooses a set of probe functions. LetQ : 2X !2R

n be a mapping on a subset of 2X into a subset of 2Rn . For example, let A;B 2 2X and Q(A);Q(B) denotesets of descriptions of points in A;B , respectively. That is,

Q(A) = f(a) : a 2 Ag ;Q(B) = f(b) : b 2 Bg :The expression A B reads A is descriptively near B . Similarly, A B reads A is descriptively far from B . Thedescriptive proximity of A and B is dened by

A B , Q(cl(A)) Q(cl(B)) 6= ;:The descriptive intersection \ of A and B is dened by

A \ B = fx 2 A [B : Q(A) Q(B)g :That is, x 2 A [B is in A \ B , provided (x) = (a) = (b) for some a 2 A; b 2 B . Observe that A and Bcan be disjoint and yet A \ B can be nonempty. The descriptive proximity relation is dened by

=(A;B) 2 2X 2X : cl(A) \ cl(B) 6= ;

:

Whenever sets A and B have no points with matching descriptions, the sets are descriptively far from each other(denoted by A B ).The binary relation is a descriptive EF-proximity, provided the following axioms are satised for A;B;E X .


dEF.1 If the set A is descriptively close to B , then B is descriptively close to A .dEF.2 A [B is descriptively close to E , if and only if, at least one of the sets A or B is descriptively close to E .dEF.3 Two points x; y 2 X are descriptively close, if and only if, the description of x matches the description of

y .dEF.4 All nonempty sets are descriptively far from the empty set ; .dEF.5 For any two sets A and B which are descriptively far from each other, there exists C;D 2 2X , C [D = X

, such that A is descriptively far from C and B is descriptively far from D (Descriptive Efremovi axiom).

The pair (X; ) is called a descriptive proximity space.

3.7 Proximal relator spacesA relator is a nonvoid family of relations R on a nonempty set X [72]. The pair (X;R) (also denoted X(R) ) iscalled a relator space. Relator spaces are natural generalizations of ordered sets and uniform spaces[73][74]}. With theintroduction of a family of proximity relationsR onX , we obtain a proximal relator space (X;R) . For simplicity,we consider only two proximity relations, namely, the Efremovi proximity [8] and the descriptive proximity indening the descriptive relatorR [53][48]. The pair (X;R) is called a proximal relator space [49]. In this work, Xdenotes a metric topological space that is endowed with the relations in a proximal relator. With the introduction of(X;R) , the traditional closure of a subset (e.g., [9][7]) can be compared with the more recent descriptive closureof a subset.In a proximal relator space X , the descriptive closure of a set A (denoted by cl(A) ) is dened by

cl(A) = fx 2 X : (x)Q(cl(A))g :That is, x 2 X is in the descriptive closure of A , provided the closure of (x) and the closure of Q(cl(A)) have atleast one element in common.

Theorem 2 [50] The descriptive closure of any set A in the descriptive EF-proximity space (X;R) is the set ofpoints x 2 X that are descriptively close to A .

Theorem 3 [50] Kuratowski closure of a setA is a subset of the descriptive closure ofA in a descriptive EF-proximityspace.

Theorem 4 [49] Let (X;R) be a proximal relator space, A X . Then cl(A) cl(A) .

Proof Let (x) 2 Q(X n cl(A)) such that (x) = (a) for some a 2 clA . Consequently, (x) 2 Q(cl(A)) .Hence, cl(A) cl(A)

In a proximal relator space, EF-proximity leads to the following results for descriptive proximity .

Theorem 5 [49] Let (X;R) be a proximal relator space, A;B;C X . Then

1 A B implies A B .

2 (A [B) C implies (A [B) C .

3 clA clB implies clA clB .

Proof

3.8. DESCRIPTIVE -NEIGHBOURHOODS 25

1 A B , A \B 6= ; . For x 2 A \B;(x) 2 Q(A) and (x) 2 Q(B) . Consequently, A B .

1 ) 2

3 clA clB implies that clA and clA have at least one point in common. Hence, 1 o ) 3o .

3.8 Descriptive -neighbourhoods

X

BE 1

E 2

A

X \E 2

Figure 6. Example depicting -neighbourhoods

In a pseudometric proximal relator space X , the neighbourhood of a point x 2 X (denoted by Nx;" ), for " > 0 , isdened by

Nx;" = fy 2 X : d(x; y) < "g :

The interior of a set A (denoted by int(A) ) and boundary of A (denoted by bdy(A) ) in a proximal relator space Xare dened by

int(A) = fx 2 X : Nx;" Ag :

bdy(A) = cl(A) n int(A):


A set A has a natural strong inclusion in a set B associated with [5][6]} (denoted by A B ), provided A intB, i.e., A X n intB ( A is far from the complement of intB ). Correspondingly, a set A has a descriptive stronginclusion in a set B associated with (denoted by A B ), provided Q(A) Q(intB) , i.e., A X n intB (Q(A) is far from the complement of intB ).Let be a descriptive -neighbourhood relation dened by

=(A;B) 2 2X 2X : Q(A) Q(intB) :

That is,A B , provided the description of each a 2 A is contained in the set of descriptions of the points b 2 intB. Now observe that any A;B in the proximal relator space X such that A B have disjoint -neighbourhoods,i.e.,

A B , A E1; B E2; for some E1; E2 X (See Fig. 6).

Theorem 6 [50] Any two sets descriptively far from each other belong to disjoint descriptive -neighbourhoods ina descriptive proximity space X .

A consideration of strong containment of a nonempty set in another set leads to the study of hit-and-miss topologiesand the Wijsman topology[2].

3.9 Tolerance near setsLet " be a real number greater than zero. In the study of sets that are proximally near within some tolerance, the setof proximity relationsR is augmented with a pseudometric tolerance proximity relation (denoted by ;" ) denedby

D(A;B) = inf fd((a);(a)) : (a) 2 Q(A);(a) 2 Q(B)g ;d((a);(a)) =

Xni=1ji(a) i(b)j;

;" =(A;B) 2 2X 2X : jD(cl(A); cl(B))j < " :

LetR;" = R [f;"g . In other words, a nonempty set equipped with the proximal relatorR;" has underlyingstructure provided by the proximal relatorR and provides a basis for the study of tolerance near sets in X that arenear within some tolerance. Sets A;B in a descriptive pseudometric proximal relator space (X;R;") are tolerancenear sets (i.e., A ;" B ), provided

D(A;B) < ":

3.10 Tolerance classes and preclassesRelations with the same formal properties as similarity relations of sensations considered by Poincar[62] are nowadays,after Zeeman[83], called tolerance relations. A tolerance on a set O is a relation O O that is reexive andsymmetric. In algebra, the term tolerance relation is also used in a narrow sense to denote reexive and symmetricrelations dened on universes of algebras that are also compatible with operations of a given algebra, i.e., they aregeneralizations of congruence relations (see e.g.,[12]). In referring to such relations, the term algebraic tolerance orthe term algebraic tolerance relation is used. Transitive tolerance relations are equivalence relations. A set O togetherwith a tolerance is called a tolerance space (denoted (O; ) ). A set A O is a -preclass (or briey preclasswhen is understood) if and only if for any x; y 2 A , (x; y) 2 .The family of all preclasses of a tolerance space is naturally ordered by set inclusion and preclasses that are maximalwith respect to set inclusion are called -classes or just classes, when is understood. The family of all classes ofthe space (O; ) is particularly interesting and is denoted by H (O) . The family H (O) is a covering of O [58].

3.10. TOLERANCE CLASSES AND PRECLASSES 27

The work on similarity by Poincar and Zeeman presage the introduction of near sets[44][43] and research on similarityrelations, e.g.,[79]. In science and engineering, tolerance near sets are a practical application of the study of sets thatare near within some tolerance. A tolerance " 2 (0;1] is directly related to the idea of closeness or resemblance(i.e., being within some tolerance) in comparing objects. By way of application of Poincar's approach in deningvisual spaces and Zeemans approach to tolerance relations, the basic idea is to compare objects such as image patchesin the interior of digital images.

3.10.1 ExamplesSimple ExampleThe following simple example demonstrates the construction of tolerance classes from real data. Consider the 20objects in the table below with jj = 1 .

Let a tolerance relation be dened as

="= f(x; y) 2 O O : k (x) (y) k2 "gThen, setting " = 0:1 gives the following tolerance classes:

H="(O) =ffx1; x8; x10; x11g; fx1; x9; x10; x11; x14g;fx2; x7; x18; x19g;fx3; x12; x17g;fx4; x13; x20g; fx4; x18g;fx5; x6; x15; x16g; fx5; x6; x15; x20g;fx6; x13; x20gg:

Observe that each object in a tolerance class satises the condition k (x)(y) k2 " , and that almost all of theobjects appear in more than one class. Moreover, there would be twenty classes if the indiscernibility relation wasused since there are no two objects with matching descriptions.Image Processing Example

Figure 7. Example of images that are near each other. (a) and (b) Images from the freely available LeavesDataset (see, e.g.,www.vision.caltech.edu/archive.html).

The following example provides an example based on digital images. Let a subimage be dened as a small subset ofpixels belonging to a digital image such that the pixels contained in the subimage form a square. Then, let the sets Xand Y respectively represent the subimages obtained from two dierent images, and let O = fX [ Y g . Finally, letthe description of an object be given by the Green component in the RGB color model. The next step is to nd all


the tolerance classes using the tolerance relation dened in the previous example. Using this information, toleranceclasses can be formed containing objects that have similar (within some small " ) values for the Green componentin the RGB colour model. Furthermore, images that are near (similar) to each other should have tolerance classesdivided among both images (instead of a tolerance classes contained solely in one of the images). For example, thegure accompanying this example shows a subset of the tolerance classes obtained from two leaf images. In thisgure, each tolerance class is assigned a separate colour. As can be seen, the two leaves share similar toleranceclasses. This example highlights a need to measure the degree of nearness of two sets.

3.11 Nearness measureLet (U;R;") denote a particular descriptive pseudometric EF-proximal relator space equipped with the proximityrelation ;" and with nonempty subsets X;Y 2 2U and with the tolerance relation=;" dened in terms of a set ofprobes and with " 2 (0;1] , where

Figure 8. Examples of degree of nearness between two sets: (a) High degree of nearness, and (b) Low degree of nearness.

';"= f(x; y) 2 U U j j(x) (y)j "g:Further, assume Z = X [ Y and let H;"(Z) denote the family of all classes in the space (Z;';") .Let A X;B Y . The distance D

tNM: 2U 2U :! [0;1] is dened by

DtNM

(X;Y ) =

(1 tNM(A;B); if X and Y are not empty;1; if X or Y is empty;

where

tNM(A;B) =

XC2H;" (Z)

jCj!1

X

C2H;" (Z)jCjmin(jC \Aj; j[C \Bj)max(jC \Aj; jC \Bj) :

The details concerning tNM are given in[14][16][17]. The idea behind tNM is that sets that are similar should havea similar number of objects in each tolerance class. Thus, for each tolerance class obtained from the covering ofZ = X [Y , tNM counts the number of objects that belong to X and Y and takes the ratio (as a proper fraction) oftheir cardinalities. Furthermore, each ratio is weighted by the total size of the tolerance class (thus giving importanceto the larger classes) and the nal result is normalized by dividing by the sum of all the cardinalities. The range oftNM is in the interval [0,1], where a value of 1 is obtained if the sets are equivalent (based on object descriptions)and a value of 0 is obtained if they have no descriptions in common.As an example of the degree of nearness between two sets, consider gure below in which each image consists of twosets of objects, X and Y . Each colour in the gures corresponds to a set where all the objects in the class share the

3.12. NEAR SET EVALUATION AND RECOGNITION (NEAR) SYSTEM 29

same description. The idea behind tNM is that the nearness of sets in a perceptual system is based on the cardinalityof tolerance classes that they share. Thus, the sets in left side of the gure are closer (more near) to each other interms of their descriptions than the sets in right side of the gure.

3.12 Near set evaluation and recognition (NEAR) system

Figure 9. NEAR system GUI.

The Near set Evaluation and Recognition (NEAR) system, is a system developed to demonstrate practical applicationsof near set theory to the problems of image segmentation evaluation and image correspondence. It was motivatedby a need for a freely available software tool that can provide results for research and to generate interest in nearset theory. The system implements a Multiple Document Interface (MDI) where each separate processing task isperformed in its own child frame. The objects (in the near set sense) in this system are subimages of the images beingprocessed and the probe functions (features) are image processing functions dened on the subimages. The systemwas written in C++ and was designed to facilitate the addition of new processing tasks and probe functions. Currently,the system performs six major tasks, namely, displaying equivalence and tolerance classes for an image, performingsegmentation evaluation, measuring the nearness of two images, performing Content Based Image Retrieval (CBIR),and displaying the output of processing an image using a specic probe function.

3.13 Proximity SystemThe Proximity System is an application developed to demonstrate descriptive-based topological approaches to near-ness and proximity within the context of digital image analysis. The Proximity System grew out of the work of S.Naimpally and J. Peters on Topological Spaces. The Proximity System was written in Java and is intended to run intwo dierent operating environments, namely on Android smartphones and tablets, as well as desktop platforms run-


Figure 10. The Proximity System.

ning the Java Virtual Machine. With respect to the desktop environment, the Proximity System is a cross-platformJava application for Windows, OSX, and Linux systems, which has been tested on Windows 7 and Debian Linuxusing the Sun Java 6 Runtime. In terms of the implementation of the theoretical approaches, both the Android andthe desktop based applications use the same back-end libraries to perform the description-based calculations, wherethe only dierences are the user interface and the Android version has less available features due to restrictions onsystem resources.

3.14 See also Alternative set theory Category:Mathematical relations Category:Topology Feature vector Proximity space Rough set Topology

3.15 Notes1. ^ J.R. Isbell observed that the notions near and far are important in a uniform space. Sets A;B are far

(uniformaly distal), provided the fA;Bg is a discrete collection. A nonempty set U is a uniform neighbour-hood of a set A , provided the complement of U is far from U . See, 33 in [23]

2. ^ The intuition that led to the discovery of descriptively near sets is given in Pawlak, Z.;Peters, J.F. (2002,2007) Jak blisko (How Near)". Systemy Wspomagania Decyzji I 57 (109)

3. ^ Descriptively near sets are introduced in[48]. The connections between traditional EF-proximity and descrip-tive EF-proximity are explored in [37].

3.16. REFERENCES 31

4. ^ Reminiscent of M. Pavels approach, descriptions of members of sets objects are dened relative to vectorsof values obtained from real-valued functions called probes. See, Pavel, M. (1993). Fundamentals of patternrecognition. 2nd ed. New York: Marcel Dekker, for the introduction of probe functions considered in thecontext of image registration.

5. ^ A non-spatial view of near sets appears in, C.J. Mozzochi, M.S. Gagrat, and S.A. Naimpally, Symmetricgeneralized topological structures, Exposition Press, Hicksville, NY, 1976., and, more recently, nearness ofdisjoint sets X and Y based on resemblance between pairs of elements x 2 X; y 2 Y (i.e. x and y havesimilar feature vectors (x);(y) and the norm k (x) (y) kp< " ) See, e.g.,[43][42][53].

6. ^ The basic facts about closure of a set were rst pointed out by M. Frchet in[11], and elaborated by B. Knasterand C. Kuratowski in[25].

7. ^Observe that up to the 1970s, proximity meant EF-proximity, since this is the one that was studied intensively.The pre-1970 work on proximity spaces is exemplied by the series of papers by J. M. Smirnov during therst half of the 1950s[68][67][69][70], culminating in the compendious collection of results by S.A. Naimpally andB.D. Warrack[34]. But in view of later developments, there is a need to distinguish between various proximities.A basic proximity or ech-proximity was introduced by E. ech during the late 1930s (see 25 A.1, pp. 439-440 in [78]). The conditions for the non-symmetric case for a proximity were introduced by S. Leader[28] andfor the symmetric case by M.W. Lodato[29][30][31].

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3.17. FURTHER READING 35

81. ^ Weil, A. (1938), Sur les espaces structure uniforme et sur la topologie gnrale, Actualits scientiqueet industrielles, Paris: Harmann & cie Missing or empty |title= (help)

82. ^ Wolski, M. (2010). Perception and classication. A note on near sets and rough sets. Fundamenta Infor-maticae 101: 143155.

83. ^ a b Zeeman, E. C. (1962), The topology of the brain and visual perception, in Fort, Jr., M. K., Topologyof 3-Manifolds and Related Topics, University of Georgia Institute Conference Proceedings (1962): Prentice-Hall, pp. 240256 Missing or empty |title= (help)

3.17 Further reading Naimpally, S. A.; Peters, J. F. (2013). Topology with Applications. Topological Spaces via Near and Far.

World Scientic Publishing . Co. Pte. Ltd. ISBN 978-981-4407-65-6. Naimpally, S. A.; Peters, J. F.; Wolski, M. (2013), "Near Set Theory and Applications", Mathematics in

Computer Science 7 (1), Berlin: Springer Missing or empty |title= (help) Peters, J. F. (2014), "Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces", Intelligent

Systems Reference Library 63, Berlin: Springer Missing or empty |title= (help) Henry, C. J.; Peters, J. F. (2012), "Near set evaluation and recognition (NEAR) system V3.0", UM CI Labo-

ratory Technical Report No. TR-2009-015, Computational Intelligence Laboratory, University of ManitobaMissing or empty |title= (help)

Concilio, A. Di (2014). Proximity: A powerful tool in extension theory, function spaces, hyperspaces, booleanalgebras and point-free geometry. Computational Intelligence Laboratory, University of Manitoba. UM CILaboratory Technical Report No. TR-2009-021.

Peters, J. F.; Naimpally, S. A. (2012). Applications of near sets (PDF).Notices of the AmericanMathematicalSociety 59 (4): 536542. CiteSeerX: 10 .1 .1 .371 .7903.

Chapter 4

Partial equivalence relation

In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restrictedequivalence relation) R on a set X is a relation that is symmetric and transitive. In other words, it holds for alla; b; c 2 X that:

1. if aRb , then bRa (symmetry)

2. if aRb and bRc , then aRc (transitivity)

If R is also reexive, then R is an equivalence relation.

4.1 Properties and applicationsIn a set-theoretic context, there is a simple structure to the general PER R on X : it is an equivalence relation on thesubset Y = fx 2 XjxRxg X . ( Y is the subset of X such that in the complement of Y ( X n Y ) no element isrelated by R to any other.) By construction, R is reexive on Y and therefore an equivalence relation on Y . Noticethat R is actually only true on elements of Y : if xRy , then yRx by symmetry, so xRx and yRy by transitivity.Conversely, given a subset Y of X, any equivalence relation on Y is automatically a PER on X.PERs are therefore used mainly in computer science, type theory and constructive mathematics, particularly to denesetoids, sometimes called partial setoids. The action of forming one from a type and a PER is analogous to theoperations of subset and quotient in classical set-theoretic mathematics.Every partial equivalence relation is a difunctional relation, but the converse does not hold.The algebraic notion of congruence can also be generalized to partial equivalences, yielding the notion of subcongruence,i.e. a homomorphic relation that is symmetric and transitive, but not necessarily reexive.[1]

4.2 ExamplesA simple example of a PER that is not an equivalence relation is the empty relation R = ; (unless X = ; , in whichcase the empty relation is an equivalence relation (and is the only relation on X )).

4.2.1 Euclidean parallelism

In the Euclidean plane, two lines m and n are parallel lines when m n = . The symmetry of this relation is obviousand the transitivity can be proven in the Euclidean plane, thus Euclidean parallelism is a partial equivalence relation.Nevertheless, mathematicians developing ane geometry prefer the facility of an equivalence relation and thereforesometimes revise the denition of parallelism to allow a line to be parallel to itself, making the new relation of aneparallelism that is a reexive relation.

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4.3. REFERENCES 37

4.2.2 Kernels of partial functionsFor another example of a PER, consider a set A and a partial function f that is dened on some elements of A butnot all. Then the relation dened by

x y if and only if f is dened at x , f is dened at y , and f(x) = f(y)

is a partial equivalence relation but not an equivalence relation. It possesses the symmetry and transitivity properties,but it is not reexive since if f(x) is not dened then x 6 x in fact, for such an x there is no y 2 A such thatx y . (It follows immediately that the subset of A for which is an equivalence relation is precisely the subset onwhich f is dened.)

4.2.3 Functions respecting equivalence relationsLet X and Y be sets equipped with equivalence relations (or PERs) X ;Y . For f; g : X ! Y , dene f g tomean:

8x0 x1; x0 X x1 ) f(x0) Y g(x1)

then f f means that f induces a well-dened function of the quotients X/ X ! Y / Y . Thus, the PER captures both the idea of denedness on the quotients and of two functions inducing the same function on thequotient.

4.3 References[1] J. Lambek (1996). The Buttery and the Serpent. In Aldo Ursini, Paulo Agliano. Logic and Algebra. CRC Press. pp.

161180. ISBN 978-0-8247-9606-8.

Mitchell, John C. Foundations of programming languages. MIT Press, 1996. D.S. Scott. Data types as lattices. SIAM Journ. Comput., 3:523-587, 1976.

4.4 See also Equivalence relation Binary relation

Chapter 5

Partial function

Not to be confused with partial function of a multilinear map or the mathematical concept of a piecewise function.

In mathematics, a partial function from X to Y (written as f: X Y) is a function f: X Y, for some subset Xof X. It generalizes the concept of a function f: X Y by not forcing f to map every element of X to an elementof Y (only some subset X of X). If X = X, then f is called a total function and is equivalent to a function. Partialfunctions are often used when the exact domain, X, is not known (e.g. many functions in computability theory).Specically, we will say that for any x X, either:

f(x) = y Y (it is dened as a single element in Y) or f(x) is undened.

For example we can consider the square root function restricted to the integers

g : Z! Zg(n) =

pn:

Thus g(n) is only dened for n that are perfect squares (i.e., 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undened.

5.1 Basic conceptsThere are two distinct meanings in current mathematical usage for the notion of the domain of a partial function.Most mathematicians, including recursion theorists, use the term domain of f" for the set of all values x such thatf(x) is dened (X' above). But some, particularly category theorists, consider the domain of a partial function f:X Y to be X, and refer to X' as the domain of denition. Similarly, the term range can refer to either the codomain orthe image of a function.Occasionally, a partial function with domain X and codomain Y is written as f: X Y, using an arrow with verticalstroke.A partial function is said to be injective or surjective when the total function given by the restriction of the partialfunction to its domain of denition is. A partial function may be both injective and surjective.Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partialfunction which is injective.[1]

An injective partial function may be inverted to an injective partial function, and a partial function which is bothinjective and surjective has an injective function as inverse. Furthermore, a total function which is injective may beinverted to an injective partial function.The notion of transformation can be generalized to partial functions as well. A partial transformation is a functionf: A B, where both A and B are subsets of some set X.[2]

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5.2. TOTAL FUNCTION 39

5.2 Total functionTotal function is a synonym for function. The use of the prex total is to suggest that it is a special case over a largerset X of a partial function over a subset of X. For example, when considering the operation of morphism compositionin Concrete Categories, the composition operation : Hom(C) Hom(C) ! Hom(C) is a total function if andonly if Ob(C) has one element. The reason for this is that two morphisms f : X ! Y and g : U ! V can only becomposed as g f if Y = U , that is, the codomain of f must equal the domain of g .

5.3 Discussion and examplesThe rst diagram above represents a partial function that is not a total function since the element 1 in the left-hand setis not associated with anything in the right-hand set. Whereas, the second diagram represents a total function sinceevery element on the left-hand set is associated with exactly one element in the right hand set.

5.3.1 Natural logarithm

Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positivereal is not a real number, so the natural logarithm function doesn't associate any real number in the codomain withany non-positive real number in the domain. Therefore, the natural logarithm function is not a total function whenviewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only includethe positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals),then the natural logarithm is a total function.

5.3.2 Subtraction of natural numbers

Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:

f : N N! N

f(x; y) = x y:It is dened only when x y .

5.3.3 Bottom element

In denotational semantics a partial function is considered as returning the bottom element when it is undened.In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEEoating point standard denes a not-a-number value which is returned when a oating point operatio

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