The classification of fuzzy subgroups of some finite non-cyclic abelian p- groups of rank 3, with emphasis on the number of distinct fuzzy subgroups

**Authors:**Appiah, Isaac Kwadwo**Date:**2021-03**Subjects:**Fuzzy sets , Commutative algebra**Language:**English**Type:**Doctoral theses , text**Identifier:**http://hdl.handle.net/10353/20783 , vital:46563**Description:**In [6] and [7] we classi_ed fuzzy subgroups of some rank-3 abelian groups of the form G = Zpn + Zp + Zp for any _xed prime integer p and any positive integer n, using the natural equivalence relation de_ned in [40]. In this thesis, we extend our classi_cation of fuzzy subgroups in [6] to the group G = Zpn + Zpm + Zp for any _xed prime integer p; m = 2 and any positive integer n using the same natural equivalence relation studied in [40]. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups of G for any n;m _ 2 and (iii) distinct fuzzy subgroups for m = 2 and n _ 2. We have also developed user-friendly polynomial formulae for the number of (iv) subgroups, (v) maximal chains for the group G = Zpn + Zpm for any n;m _ 2; any _xed prime positive integer p and (vi) distinct fuzzy subgroups of Zpn + Zpm for m equal to 2 and 3, and n _ 2 and provided their proofs. , Thesis (PhD) -- Faculty of Science and Agriculture, 2021**Full Text:****Date Issued:**2021-03

**Authors:**Appiah, Isaac Kwadwo**Date:**2021-03**Subjects:**Fuzzy sets , Commutative algebra**Language:**English**Type:**Doctoral theses , text**Identifier:**http://hdl.handle.net/10353/20783 , vital:46563**Description:**In [6] and [7] we classi_ed fuzzy subgroups of some rank-3 abelian groups of the form G = Zpn + Zp + Zp for any _xed prime integer p and any positive integer n, using the natural equivalence relation de_ned in [40]. In this thesis, we extend our classi_cation of fuzzy subgroups in [6] to the group G = Zpn + Zpm + Zp for any _xed prime integer p; m = 2 and any positive integer n using the same natural equivalence relation studied in [40]. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups of G for any n;m _ 2 and (iii) distinct fuzzy subgroups for m = 2 and n _ 2. We have also developed user-friendly polynomial formulae for the number of (iv) subgroups, (v) maximal chains for the group G = Zpn + Zpm for any n;m _ 2; any _xed prime positive integer p and (vi) distinct fuzzy subgroups of Zpn + Zpm for m equal to 2 and 3, and n _ 2 and provided their proofs. , Thesis (PhD) -- Faculty of Science and Agriculture, 2021**Full Text:****Date Issued:**2021-03

A study of fuzzy sets and systems with applications to group theory and decision making

**Authors:**Gideon, Frednard**Date:**2006**Subjects:**Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5417 , http://hdl.handle.net/10962/d1005231 , Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Description:**In this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections.**Full Text:****Date Issued:**2006

**Authors:**Gideon, Frednard**Date:**2006**Subjects:**Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5417 , http://hdl.handle.net/10962/d1005231 , Fuzzy sets , Fuzzy systems , Abelian groups , Decision making**Description:**In this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections.**Full Text:****Date Issued:**2006

Sobriety of crisp and fuzzy topological spaces

**Authors:**Jacot-Guillarmod, Paul**Date:**2004**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5414 , http://hdl.handle.net/10962/d1005228 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.**Full Text:****Date Issued:**2004

**Authors:**Jacot-Guillarmod, Paul**Date:**2004**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5414 , http://hdl.handle.net/10962/d1005228 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.**Full Text:****Date Issued:**2004

Finite fuzzy sets, keychains and their applications

**Authors:**Mahlasela, Zuko**Date:**2009**Subjects:**Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5406 , http://hdl.handle.net/10962/d1005220 , Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models**Description:**The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.**Full Text:****Date Issued:**2009

**Authors:**Mahlasela, Zuko**Date:**2009**Subjects:**Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5406 , http://hdl.handle.net/10962/d1005220 , Fuzzy sets , Finite groups , Lattice theory , Economics -- Mathematical models**Description:**The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.**Full Text:****Date Issued:**2009

(L, M)-fuzzy topological spaces

**Authors:**Matutu, Phethiwe Precious**Date:**1992**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5410 , http://hdl.handle.net/10962/d1005224 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.**Full Text:****Date Issued:**1992

**Authors:**Matutu, Phethiwe Precious**Date:**1992**Subjects:**Topological spaces , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5410 , http://hdl.handle.net/10962/d1005224 , Topological spaces , Fuzzy sets**Description:**The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.**Full Text:****Date Issued:**1992

Real options valuation for South African nuclear waste management using a fuzzy mathematical approach

**Authors:**Montsho, Obakeng Johannes**Date:**2013 , 2013-06-06**Subjects:**Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5398 , http://hdl.handle.net/10962/d1003051 , Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa**Description:**The feasibility of capital projects in an uncertain world can be determined in several ways. One of these methods is real options valuation which arose from financial option valuation theory. On the other hand fuzzy set theory was developed as a mathematical framework to capture uncertainty in project management. The valuation of real options using fuzzy numbers represents an important refinement to determining capital projects' feasibility using the real options approach. The aim of this study is to determine whether the deferral of the decommissioning time (by a decade) of an electricity-generating nuclear plant in South Africa increases decommissioning costs. Using the fuzzy binomial approach, decommissioning costs increase when decommissioning is postponed by a decade whereas use of the fuzzy Black-Scholes approach yields the opposite result. A python code was developed to assist in the computation of fuzzy binomial trees required in our study and the results of the program are incorporated in this thesis. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in**Full Text:****Date Issued:**2013

**Authors:**Montsho, Obakeng Johannes**Date:**2013 , 2013-06-06**Subjects:**Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5398 , http://hdl.handle.net/10962/d1003051 , Fuzzy mathematics , Real options (Finance) , Fuzzy sets , Business mathematics , Radioactive waste disposal -- South Africa**Description:**The feasibility of capital projects in an uncertain world can be determined in several ways. One of these methods is real options valuation which arose from financial option valuation theory. On the other hand fuzzy set theory was developed as a mathematical framework to capture uncertainty in project management. The valuation of real options using fuzzy numbers represents an important refinement to determining capital projects' feasibility using the real options approach. The aim of this study is to determine whether the deferral of the decommissioning time (by a decade) of an electricity-generating nuclear plant in South Africa increases decommissioning costs. Using the fuzzy binomial approach, decommissioning costs increase when decommissioning is postponed by a decade whereas use of the fuzzy Black-Scholes approach yields the opposite result. A python code was developed to assist in the computation of fuzzy binomial trees required in our study and the results of the program are incorporated in this thesis. , KMBT_363 , Adobe Acrobat 9.54 Paper Capture Plug-in**Full Text:****Date Issued:**2013

The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes

**Authors:**Ndiweni, Odilo**Date:**2007**Subjects:**Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)**Language:**English**Type:**Thesis , Masters , M Sc (Mathematics)**Identifier:**vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)**Description:**In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .**Full Text:****Date Issued:**2007

**Authors:**Ndiweni, Odilo**Date:**2007**Subjects:**Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)**Language:**English**Type:**Thesis , Masters , M Sc (Mathematics)**Identifier:**vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)**Description:**In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .**Full Text:****Date Issued:**2007

Case studies of equivalent fuzzy subgroups of finite abelian groups

**Authors:**Ngcibi, Sakhile L**Date:**2002**Subjects:**Abelian groups , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5401 , http://hdl.handle.net/10962/d1005215 , Abelian groups , Fuzzy sets**Description:**The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.**Full Text:****Date Issued:**2002

**Authors:**Ngcibi, Sakhile L**Date:**2002**Subjects:**Abelian groups , Fuzzy sets**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5401 , http://hdl.handle.net/10962/d1005215 , Abelian groups , Fuzzy sets**Description:**The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.**Full Text:****Date Issued:**2002

Methods for designing and optimizing fuzzy controllers

**Authors:**Swartz, Andre Michael**Date:**2000**Subjects:**Fuzzy sets , Fuzzy systems , Automatic control**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5412 , http://hdl.handle.net/10962/d1005226 , Fuzzy sets , Fuzzy systems , Automatic control**Description:**We start by discussing fuzzy sets and the algebra of fuzzy sets. We consider some properties of fuzzy modeling tools. This is followed by considering the Mamdani and Sugeno models for designing fuzzy controllers. Various methods for using sets of data for desining controllers are discussed. This is followed by a chapter illustrating the use of genetic algorithms in designing and optimizing fuzzy controllers.Finally we look at some previous applications of fuzzy control in telecommunication networks, and illustrate a simple application that was developed as part of the present work.**Full Text:****Date Issued:**2000

**Authors:**Swartz, Andre Michael**Date:**2000**Subjects:**Fuzzy sets , Fuzzy systems , Automatic control**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5412 , http://hdl.handle.net/10962/d1005226 , Fuzzy sets , Fuzzy systems , Automatic control**Description:**We start by discussing fuzzy sets and the algebra of fuzzy sets. We consider some properties of fuzzy modeling tools. This is followed by considering the Mamdani and Sugeno models for designing fuzzy controllers. Various methods for using sets of data for desining controllers are discussed. This is followed by a chapter illustrating the use of genetic algorithms in designing and optimizing fuzzy controllers.Finally we look at some previous applications of fuzzy control in telecommunication networks, and illustrate a simple application that was developed as part of the present work.**Full Text:****Date Issued:**2000

The principle of inclusion-exclusion and möbius function as counting techniques in finite fuzzy subsets

**Authors:**Talwanga, Matiki**Date:**2009**Subjects:**Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5413 , http://hdl.handle.net/10962/d1005227 , Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function**Description:**The broad goal in this thesis is to enumerate elements and fuzzy subsets of a finite set enjoying some useful properties through the well-known counting technique of the principle of inclusion-exclusion. We consider the set of membership values to be finite and uniformly spaced in the real unit interval. Further we define an equivalence relation with regards to the cardinalities of fuzzy subsets providing the Möbius function and Möbius inversion in that context.**Full Text:****Date Issued:**2009

**Authors:**Talwanga, Matiki**Date:**2009**Subjects:**Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function**Language:**English**Type:**Thesis , Masters , MSc**Identifier:**vital:5413 , http://hdl.handle.net/10962/d1005227 , Fuzzy logic , Fuzzy sets , Fuzzy systems , Möbius function**Description:**The broad goal in this thesis is to enumerate elements and fuzzy subsets of a finite set enjoying some useful properties through the well-known counting technique of the principle of inclusion-exclusion. We consider the set of membership values to be finite and uniformly spaced in the real unit interval. Further we define an equivalence relation with regards to the cardinalities of fuzzy subsets providing the Möbius function and Möbius inversion in that context.**Full Text:****Date Issued:**2009

Counting of finite fuzzy subsets with applications to fuzzy recognition and selection strategies

**Authors:**Talwanga, Matiki**Date:**2015**Subjects:**Möbius transformations , Fuzzy sets , Functions, Zeta , Partitions (Mathematics)**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5431 , http://hdl.handle.net/10962/d1018186**Description:**The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place.**Full Text:****Date Issued:**2015

**Authors:**Talwanga, Matiki**Date:**2015**Subjects:**Möbius transformations , Fuzzy sets , Functions, Zeta , Partitions (Mathematics)**Language:**English**Type:**Thesis , Doctoral , PhD**Identifier:**vital:5431 , http://hdl.handle.net/10962/d1018186**Description:**The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place.**Full Text:****Date Issued:**2015

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