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Equivalence relation (X) Smarandache Collections (X)

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 Book Id: WPLBN0002097645 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Literature, Education, Authors Community, Mathematics ► Abstract Full Text Search Details...ions. In the first section we just give the basic concepts or notion like equivalence relation greatest common divisor etc. Second section is devote... ...e first section we just give the basic concepts or notion like equivalence relation greatest common divisor etc. Second section is devoted to giving ... ...epts The main aim of this section is to introduce the basic concepts of equivalence relation, equivalence class and introduce some number theoreti... ...ain aim of this section is to introduce the basic concepts of equivalence relation, equivalence class and introduce some number theoretic results use... ...= c -1 ac. We denote a conjugate to b by a ~ b and we shall refer to this relation as conjugacy relation on G. 9 DEFINITION 1.2.4: Let G be a... ...properties of them. DEFINITION 1.3.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are cal... ...ive if for a, b, c in A, aRb and bRc imply aRc. A relation R on A is an equivalence relation if R is reflexive, symmetric and transitive. In this ... ... symmetric and transitive. In this case [a] = {b ∈ A / aRb} is called the equivalence class of a for any a ∈ A. DEFINITION 1.3.2: A relation R o... ... This problem leads to the definition of Smarandache right (left) coset equivalence sets in a loop L related to a subgroup A of L. In chapter 5 se...
 Book Id: WPLBN0002097108 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Description Details...The aim of this book is two fold. At the outset the book gives most of the available literature about Fuzzy Relational Equations (FREs) and its properties for there is no book that solely caters to FREs and its applications. Though we have a comprehensive bibliography, we do not promise to give all the possible available literature... Full Text Search Details... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS HEXI... ...OSOPHIC RELATIONAL MAPS HEXIS Church Rock 2004 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasan... ... i c a t i o n 6 P r e f a c e 7 Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES 1.1 Fuzzy Rel... ... Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility relations and composition o... ...In this section we just recollect the properties of fuzzy relations like, fuzzy equivalence relation, fuzzy compatibility relations, fuzzy ordering ... ...more about these concepts please refer [43]. Now we proceed on to define fuzzy equivalence relation. A crisp binary relation R(X, X) that is reflex... ...inary relation R(X, X) that is reflexive, symmetric and transitive is called an equivalence relation. For each element x in X, we can define a crisp... ...sp set A x , which contains all the elements of X that are related to x, by the equivalence relation. A x = {y ⏐(x, y) ∈ R (X, X)} A x is cl... ...ed to any element of X not included in A x . This set A x is referred to an as equivalence class 17 of R (X, X) with respect to x. The members...
 Book Id: WPLBN0002097105 ► Abstract Full Text Search Details... W. B. VASANTHA KANDASAMY FLORENTIN SMARANDACHE FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS HEXI... ...OSOPHIC RELATIONAL MAPS HEXIS Church Rock 2004 1 FUZZY RELATIONAL MAPS AND NEUTROSOPHIC RELATIONAL MAPS W. B. Vasan... ... i c a t i o n 6 P r e f a c e 7 Chapter One FUZZY RELATIONAL EQUATIONS: BASIC CONCEPTS AND PROPERTIES 1.1 Fuzzy Rel... ... Fuzzy Relational Equations and their properties 10 1.2 Properties of Fuzzy Relations 16 1.3 Fuzzy compatibility relations and composition o... ...In this section we just recollect the properties of fuzzy relations like, fuzzy equivalence relation, fuzzy compatibility relations, fuzzy ordering ... ...more about these concepts please refer [43]. Now we proceed on to define fuzzy equivalence relation. A crisp binary relation R(X, X) that is reflex... ...inary relation R(X, X) that is reflexive, symmetric and transitive is called an equivalence relation. For each element x in X, we can define a crisp... ...sp set A x , which contains all the elements of X that are related to x, by the equivalence relation. A x = {y ⏐(x, y) ∈ R (X, X)} A x is cl... ...ed to any element of X not included in A x . This set A x is referred to an as equivalence class 17 of R (X, X) with respect to x. The members...
 Book Id: WPLBN0002097651 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...× µ ) (x, y) = min {λ(x), µ (y)} for every (x, y) ∈ X × Y. A fuzzy binary relation R λ on a set X is defined as a fuzzy subset of X × X. The co... ...X is defined as a fuzzy subset of X × X. The composition of two fuzzy relations R λ and R µ is defined by (R λ o R µ )(x, y) = sup X t∈ {min R... ... y)}, for every x, y ∈ X. DEFINITION 1.1.12: Let R λ be a fuzzy binary relation on a set X. A fuzzy subset µ of the set X is said to be a pre cla... ... {µ (x), µ (y) } ≤ R λ (x, y) for every x, y ∈ X. 11 A fuzzy binary relation R λ on a set X is said to be a similarity relation on the set X i... ....2: A relation ρ on an R-module M is called a congruence on M if it is an equivalence relation on M such that (a, b) ∈ ρ and (c, d) ∈ ρ imply that (a... ...uzzy relation α on M [i.e. a mapping α : M × M → [0,1]] is called a fuzzy equivalence relation if i. z , y , x all for ) z , y ( ) x , x ( sup M ... ...α α ∈ ≥ for all x, y in M (fuzzy transitive). DEFINITION 1.8.5: A fuzzy equivalence relation α on an R-module M is called a fuzzy congruence if α ... ... [] [] ) y , z ( ), z , x ( Min sup M z µ µ ∈ α α . Thus α µ is a fuzzy equivalence relation on M. Now α µ (x + u, y + ν) = µ (x + u – ν – y... ... generates the half groupoid P. Now we introduce the notions of fuzzy equivalence relation as given by [144] and proceed on to recall the defini...
 Book Id: WPLBN0002097090 Subjects: Non Fiction, Philosophy, Smarandache Collections ► Abstract Full Text Search Details...lly, chapter four makes an introduction to neutrosophic logic, where the relationship between it and the other logics will be clear. 4 As alway... ...tions Section 2 The Law of the Excluded Middle Section 3 Logical Equivalence Section 4 Well-Formed Formulas or WFFs Section 5 An A... ...dent is false have a value of true. 14 Section 3 Logical Equivalence Definition 1.3.1: The propositions p and q are said to be l... ...to be logically equivalent if p ↔ q is a tautology. The notation for this relationship is p q. It is possible to show that two expressions... ...uth table. Examples: It is easy to verify each of the following logical equivalences ( p → q ) (¬ p \/ q ) ( p ↔ q ) ... ...pressions are taken to be true by assumption. d) There is a finite set of relations R 1 , R 2 , . . . , R n between sets of the wffs in S, called t... ... truth value of the expression is dependent on internal variables and the relationships between the features. For example, in the first expression, ... ...q could be defined as max{ p, q } and p /\ q as min { p, q }. Using the equivalence formulas (p → q) (¬p \/ q), ((p /\ q) \/ (¬p /\ ¬q)) ... ... The definitions of the connectives in table 3 are consistent with the equivalence formulas in classical logic. It is not necessary to maintain t...
 Book Id: WPLBN0002097100 ► Abstract Full Text Search Details... and applies it to the case of neutrosophic cognitive maps and neutrosophic relational maps. The last section of this chapter clearly illustrates ho... ...out this book by ‘I ’ we denote the indeterminacy of any notion / concept / relation. That is when we are not in a position to associate a relation ... ...f the neutrosophic vector space V over R. A study of these basis and its relations happens to be an interesting form of research. DEFINITION 1... ... Cognitive Maps and use rectangular neutrosophic matrices for Neutrosophic Relational Maps. 13 Chapter Two SOME BASIC RESULTS... ...ion of a graph in which people are represented by points and interpersonal relations by lines. Such relations include love, hate, communication and... ...is reachable from the other. It is easily verified that disconnection is an equivalence relation on the vertex set of D and if the equivalence class... ...onigsberg bridge problem has no solution. The following theorem which gives equivalence of three conditions is left for the reader as an exercise to...
 Book Id: WPLBN0002097644 Subjects: Non-Fiction, Education, Smarandache Collections Collections: Math, Literature, Education, Algebra, Authors Community, Mathematics ► Abstract Full Text Search Details... n . (D 2n is called the dihedral group of order 2n given by the following relation, D 2n = {a, b/ a 2 = b n = 1; bab = a}. 13. Give an examp... ...ces used in this book. DEFINITION 1.2.1: Let A and B be non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to B are c... ...lation R from A to B is a subset of A × B. Relations from A to B are called relations on A, for short, if (a, b) ∈ R then we write aRb and say that ... ...ions on A, for short, if (a, b) ∈ R then we write aRb and say that 'a is in relation R to b’. Also if a is not in relation R to b, we write b R a / ... ...itive if for a, b, c in A; aRb and bRc imply aRc. A relation R on A is an equivalence relation if R is reflexive, symmetric and transitive. In the... ...ve, symmetric and transitive. In the case [a] = {b ∈ A| aRb}, is called the equivalence class of a for any a ∈ A. DEFINITION 1.2.2: A relation R o... ... DEFINITION (MONICO, CHRIS): A congruence relation on a semiring S is an equivalence relation ~ that also satisfies        + + + + ⇒ c x... ...ongruence relation) ~ if we have a S-subsemiring A of S such that '~' is an equivalence relation that also satisfies        + + + + ⇒ ... ...61, 66, 91, 95-96, 100 Distributive law, 19, 39 Division ring, 19 E Equivalence class, 12 Equivalence relation, 12, 51, 80 Extension field...
 Book Id: WPLBN0002097095 ► Abstract Full Text Search Details... manifolds with higher dimension are inspired by the Riemann surfaces. The relation of maps with Riemann surfaces has been known in 80s in the last ce... ...etries, maps and the semi-arc automorphism group of a graph. Preface ii A relation for maps and Smarandache manifolds (abbreviated s-manifolds) and a... ...oted maps are established in this chapter. The last section determines the relation of the number of embeddings and rooted maps of a graph on genus. A... ...me for enumerating maps underlying a graph.........................17 §4 A relation among the total embeddings and rooted maps of a graph on genus20 4... ...pty boundary and to be analytic functions. Whence, we have the following relation: {Riemann Sufaces}⊂{Klein surfaces}. The upper half planeH ={z∈C|I... ...re exists an isomorphism ζ between them induced by an element ξ. Call ζ an equivalence between M 1 ,M 2 . Certainly, on an orientable surface, an equi... ...t1 2 Γ such that P ζ 1 = P 2 or P ζ 1 = P −1 2 . Proof By the deﬁnition of equivalence between maps, if κ is an equivalence between M 1 and M 2 , then... ...ducedtheconformalmappingbetween maps. Then, how can we deﬁne the conformal equivalence for maps enabling us to get the uniformiza- tion theorem of map...
 Book Id: WPLBN0002097647 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...dular lattices. DEFINITION 1.3.1: Let A and B be two non-empty sets. A relation R from A to B is a subset of A × B. Relations from A to A are ca... ...ion R from A to B is a subset of A × B. Relations from A to A are called relation on A, for short. If (a, b) ∈ R then we write aRb and say that a i... ...ion on A, for short. If (a, b) ∈ R then we write aRb and say that a is in relation R to b. Also if a is not in relation R to b we write a R / b. A r... ... a = b. R is transitive if for all a,b,c in A aRb and bRc imply aRc. A relation R on A is an equivalence relation, if R is reflexive, symmetric a... ...tive if for all a,b,c in A aRb and bRc imply aRc. A relation R on A is an equivalence relation, if R is reflexive, symmetric and transitive. In t... ...metric and transitive. In this case, [a] = {b ∈ A / aRb} is called the equivalence class of a for any a ∈ A. DEFINITION 1.3.2: A relation R on... ...exists e ∈ M such se = es = s. The reader is advised to develop Morita equivalence on semigroups with systems of local units. A ring (or semi... ...Can we ever find a ring R in which subring link relation happens to be an equivalence relation? 182. Can reals or ring of integers have pairs whi... ...es, Math. Issled, No. 66, 113-127, (1982). 58. Neklyudova V.V. Morita, Equivalence of semigroups with systems of local units, Fundam. Prikl. Math...
 Book Id: WPLBN0002097104 ► Abstract Full Text Search Details...s and their Socio-economic Conditions 5.1 Use of FRM in the study of relation between HIV/AIDS migrants and their socio-economic condition... ...he problem of HIV/AIDS Migrant Labourers 271 5.5 Linked Neutrosophic Relational Maps and its application to Migrant Problems 288 5.6 Comb... ...ctims. This study is done in Chapter IV. In chapter V we use Neutrosophic Relational Maps and we define some new neutrosophic tools like Combined Di... ...onomic terms, while ignoring that migration is 8 the result of an inter-relationship of an aggregate of several factors. As a part of our resear... ...e vital role of religious faith. Finally as several interlinking of the relations and its effect on the nodes in some cases may remain to be an in... ...s given by {A 1 , A 2 , …,A 6 } and {G 1 G 2 … G 5 }. Let us divide into equivalence classes. C 1 = {(A 1 A 2 ) (G 1 G 4 )} and C 2 = {(A 3...
 Book Id: WPLBN0002097043 Subjects: Non Fiction, Algebra, Smarandache Collections ► Abstract Full Text Search Details...el each of which assumes some economic parameter which describe the inter relations between the industries in the economy 32 under considerations... ...he collection of all polynomials of degree less than or equal to 4. So no relation among elements of V 1 and V 2 is possible. Thus we also show t... ...gebra and the linear bialgebra. Clearly the strong linear bialgebra has no relation with the linear bialgebra or a linear bialgebra has no relation ... ...d where as the strong linear bialgebra is defined over a bifield, hence no relation can ever be derived. In the similar means one cannot derive any... ...n can ever be derived. In the similar means one cannot derive any form of relation between the weak linear bialgebra and linear bialgebra. All th... ...d to define the notion of Jordan biform. The notion of Jordan form in each equivalence class of matrices under similarity and so it has been proved ...
 Book Id: WPLBN0002097103 Subjects: Non Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details... } 0I 1 /I 1 ?I 2 , {1 + } 0F 1 /F 1 ?F 2 ). 2.3.6 Material biconditional (equivalence): NL(A 1 f A 2 ) = ( ({1 + } 0T 1 /T 1 ?T 2 ) ? ({1 + ... ...: [1] Atanassov, K., Burillo, P., Bustince, H., On the intuitionistic fuzzy relations, Notes on Intuitionistic Fuzzy Sets, Vol. 1 (1995), No. 2,... ....stanford.edu/mathematics-constructive/, 1997. [5] Buhaescu, T., On an order relation between fuzzy numbers and fuzzy functions convexity, Itinera... ...d Bel Bel Bel whenever and is called a Bayesian belief function. In such case, relation (21) coincides exactly with (10) and a probability function i... ... Connec tion of the DST with the Fuzzy Set Theory can be found in [5, 64]. The relationship between experimental observations and the DST belief func... ...a. Neutrosophy considers a proposition, theory, event, concept, or entitity, in relation to its opposite, anti and that which is not , non ,a n d th... ...itional neutrosophic logical operators like (strong disjunction, implica tion, equivalence, Sheffer’s and Pierce’s connectors) and general, physics a...
 Book Id: WPLBN0002097092 ► Abstract Full Text Search Details...s-manifolds are a very restricted subclass of the 7 polyhedral surfaces. The relationship between polyhedral surfaces and Riemannian manifolds is... ... The non-Euclidean vertices introduce a sort of curvature, and this affects the relationships between s-lines. The topology of an s-manifold can all... ...lines being parallel or not parallel at different points along them, since this relationship between s-lines changes as 21 we move from point to ... ...y, to change from region to region, this does not necessarily carry over to the relationships between lines, or 26 geodesics. This is, in fact, o... ...e to S-deny all of Hilbert’s axioms, and our choice to have congruence to be an equivalence relation further reduces our ability to S-deny Hilbert’s... ...all of Hilbert’s axioms, and our choice to have congruence to be an equivalence relation further reduces our ability to S-deny Hilbert’s axioms. Mos... ...nsitive. In fact, our definition of s- congruence satisfies the properties of an equivalence relation, i.e., it is reflexive, symmetric, and transiti... ...lexive, symmetric, and transitive. Using a notion of congruence that was not an equivalence relation would complicate the study of all related issue...
 Book Id: WPLBN0002097042 ► Abstract Full Text Search Details...ith that energy being subject to the Doppler effect because of the Planck relationship E = h υ according to the factor (1- β)/√(1- β 2 ). [3] Putt... ...the electric current." A similar statement may apply to gravitons in their relation to mass. [3] A is sail area, p is light pressure, k = 2/c ... ...e traveler ages most!" The Galilean-Newtonian-Einsteinian principle of equivalence equates gravitation and acceleration fields, and the speed of ... ... field would be expected to stop completely. Finally by the principle of equivalence, the same thing would be expected to happen in an infinite acce...
 Book Id: WPLBN0002097643 Subjects: Non-Fiction, Education, Smarandache Collections ► Abstract Full Text Search Details...R + and Z ⊂ Q ⊂ R, where ' ⊂ ' denotes the containment that is ' contained ' relation. Z n = {0, 1, 2, ... , n-1} be the set of integers under mu... ... homomorphic image of a sub-semigroup of S 2 . In symbols G 1 | G 2 . The relation divides is denoted by '|'. DEFINITION: Let 1 K = (Z 1 ,... ...(z 1 , z 2 , z 3 )}, {a 1 , a 2 }, {0, 1}, δ, λ} Define the equivalence classes on Z as ~ 1 and ~ 2 . Find Z/~ 1 and Z/~ 2 . ...
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