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THE JOURNAL OF FUZZY MATHEMATICS
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The Journal of Fuzzy Mathematics

Volume 24, Number 2, June 2016

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Fuzzy Continuous Function in Fuzzy Topological Space

I. A. Noaman

Al-Baha University EL-moundq Faculty of Science and Art., P. O. Box 1988, K. S. A

M. El Sayed

Najran University Faculty of Science and Art., P. O. Box 1988, K. S. A

Abstract: In this paper, we introduced and study a new types of near fuzzy open sets in a fuzzy topological space (X,t) namely simply fuzzy open sets (for short,S-F -open sets). And also, we introduced new types of fuzzy topological operators based on S -fuzzy open sets such as, S-fuzzy closure, S-fuzzy interior, S-fuzzy boundary, S-fuzzy exterior, S-fuzzy limit points, S-fuzzy neighborhood and S-fuzzy density. Finally, we study some important properties of this sets.

Key words: Simply open set, fuzzy topological space and near continuous mappings.

Intuitionistic Fuzzy Contra ¦Á-generalized Semi Continuous Mappings

M. Jeyaraman

Department of Mathematics, Raja Dorai Singam Govt. Arts College, Sivagangai, Tamil Nadu, India. E-mail: jeya.math@gmail.com

A. Yuvarani

Department of Mathematics, NPR College of Engineering and Technology, Natham, Tamil Nadu, India. E-mail: yuvaranis@rediffmail.com

Abstract: In this paper we study the concepts of intuitionistic fuzzy contra alpha generalized semi-continuous mappings and intuitionistic fuzzy contra alpha generalized semi-irresolute mappings in intuitionistic fuzzy topological space. We also study various properties and relations between the other existing intuitionistic fuzzy contra continuous mappings.

Key words: Intuitionistic fuzzy topology, Intuitionistic fuzzy contra ¦Á-generalzied semi-continuous mapping, Intuitionistic fuzzy contra ¦Á-generalized semi-irresolute mapping.

Fuzzy Graphs and Complementary Fuzzy Graphs

John N. Mordeson

E-mail: mordes@creighton.edu

Davender S. Malik

E-mail: malik@creighton.edu

Colin D. Richards

E-mail: colinrichards@creighton.edu

Joshua A. Trebbian

Department of Mathematics. E-mail: joshuatrebbian@creighton.edu

Maureen A. Boyce

Department of Political Scienc. E-mail: maureenboyce@creighton.edu

Mark P. Byrne

Department of Computer Science. E-mail: markbyrne@creighhton.edu

Benjamin J. Cousino

Department of Political Science Creighton University Omaha, Nebraska 68178 USA. E-mail: benjamincousino@creighton.edu

Abstract: The idea of a fuzzy point and its membership to and quasi-coincidence with a fuzzy subset with respect to the standard complement have been introduced to define and study certain kinds of fuzzy topological spaces. This was generalized by using arbitrary complements with an equilibrium. In this paper, we introduce these ideas to fuzzy graph theory. We also show how the results can be applied to social network analysis, in particular to human trafficking..

Key words: Fuzzy graph, complementary fuzzy graph, fuzzy cut-vertex, fuzzy end-vertex, quasi-fuzzy graph.

An Algorithm for Solving Unbalanced Intuitionistic Fuzzy Assignment Problem Using Triangular Intuitionistic Fuzzy Number

P. Senthil Kumar

Assistant Professor, PG and Research Department of Mathematics, Jamal Mohamed College, Tiyuchirappalli-620020, Tamil Nadu, Corresponding author. E-mail: senthilsoft_5760@yahoo.com

R. Jahir Hussain

Associate Professor PG and Research Department of Mathematics, Jamal Mohamed College, Tiruchirappalli-620020, India. E-mail: hssn_jhr@yahoo.com

Abstract: In solving real life assignment problem, we often face the state of uncertainty as well as hesitation due to varies uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this paper, computationally a simple method is proposed to find the optimal solution for an unbalanced assignment problem under intuitionistic fuzzy environment. In conventional assignment problem, cost is always certain. This paper develops an approach to solve the unbalanced assignment problem where the time/cost/profit is not in deterministic numbers but imprecise ones. In this assignment problem, the elements of the cost matrix are represented by the triangular intuitionistic fuzzy numbers. The existing Ranking procedure of Varghese and Kuriakose is used to transform the unbalanced intuitionistic fuzzy assignment problem into a crisp one so that the conventional method may be applied to solve the . Finally the method is illustrated by a numerical example which is followed by graphical representation and discussion of the finding.

Key words: Intuitionistic Fuzzy Set, Triangular Intuitionistic Fuzzy Number, Unbalanced Intuitionistic Fuzzy Assignment Problem, Optimum Schedule.

A Multi-objective Solid Transportation Model with Budget and Restriction under Uncertain Environment

Amrit Das and Uttam Kuman Berav

Department of Mathematics, National Institute of Technology, Agartala, Barjala, Jirania, West Tripura-799046. E-mail:das.amrit12@gmail.com, bera_uttam@yahoo.co.in E-mail: amrishhanda83@gmail.com

Abstract: This paper investigate a multi-objective solid transportation problem with budget and restriction on the quantity of transported amount of goods under uncertain environment. Sometimes in transportation system it is observe that a least amount of goods take part in transportation, which consumes a time for transportation. In such cases the decision maker put a restriction on the amount of goods by a fix number. So that this restriction will reduce the total transportation time. Here in this paper we present basically two models, one with restriction and another without restriction in uncertain environment. All the parameters of transportation system are considered as an uncertain variable. However with the use of inverse uncertain distribution, both the proposed models are transformed to its deterministic form by taking expected value on objective functions and confidence level on the constraints. After that these crisp models are solved with the weighted mean technique and using LINGO 13.0 software. A numerical example is provided to show the application of the model.

Key words: Multi Objective Solid Transportation Problem, Uncertainty theory, Weighted Mean Technique, Budget Constraint.

Coupled Fixed Point Theorems in Modified Intuitionistic Fuzzy Metric Spaces Satisfying ¦Õ-contractive Condition with Application to Integral Equations

Bhavana Deshpande

Department of Mathematics Govt. B.S. P. G. College Jaora ( M. P.), India. E-mail:bhavnadeshpande@yahoo.com

Suresh Chouhan

Department of Mathematics Govt. Girls College Ratlam ( M. P.), India.

Shamim Ahmad Thoker

Department of Mathematics Govt. Arts and Science P. G. College Ratlam ( M. P.), India. E-mail:shamimthoker@gmail.com

Abstract: We establish coupled fixed point theorems for ¦Õ-contractive mixed monotone mappings on partially ordered non-Archimedean modified intuitionistic fuzzy metric spaces. As an application of our result we study the existence and uniqueness of the solution to a nonlinear Fredholm integral equation. We also give examples to validate our results.

Key words: Non-Archimedean modified intuitionistic fuzzy metric space, Triangular norm, Mixed monotone mapping, Complete lattice, Coupled fixed point, Contractive condition.

A Decision Making Method Based on Information Measure of Interval Valued Intuitionistic Fuzzy Soft Sets of Root Type

S. Anita Shanthi and J. Vadivel Naidu

Department of Mathematics, Annamalai University, Annamalainagar - 608002, Tamilnadu, India. E-mail:shanthi.anita@yahoo.com, mathsvel@yahoo.com

N. Thillaigovindan

Department of Mathematics, College of Natural Sciences, Arba Minch University, Arba Minch Ethiopia, Tamilnadu, India. E-mail:ihillaigovindan.natesan@gmail.com

Abstract: In this paper, we define some operators on interval valued intuitionistic fuzzy soft set of root type and establish some properties of these operators. An information measure of interval valued intuitionistic fuzzy soft set of root type is proposed. We then develop a decision making method which is based on information measure of interval valued intuitionistic fuzzy soft set of root type. Numerical examples are given to illustrate the applications of the information measure of interval valued intuitionistic fuzzy soft set of root type to decision making.

Key words: Interval valued intuitionistic fuzzy soft set of root type, complement, ?and £¿ operators, information measure, decision making technique.

Fuzzy Quasi Semiopen Sets and Connectedness between Fuzzy Sets in Fuzzy Bitopological Spaces

A. Vadivel

Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu-608002. E-mail:avmaths@gmail.com

M. Palanisamy

Department of Mathematics, Annamalai University, Annamalainagar-608002. E-mail:palaniva26@gmail.com

Abstract: In this paper we introduce and study fuzzy quasi semiopen sets, fuzzy quasi semiclosed sets, fuzzy quasi semi connectedness between fuzzy sets, fuzzy quasi semi-preopen sets, fuzzy quasi semi-pre-separated sets in fuzzy bitopological spaces.

Key words: Fuzzy quasi semiopen, fuzzy quasi semi connectedness between fuzzy sets, pairwise fuzzy semi connected between the fuzzy sets, fuzzy quasi semi-preopen sets and fuzzy quasi semi-pre-separated sets.

Upper (Lower) Contra-continuous Intuitionistic Fuzzy Multifunctions

Kush Bohre

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, (M. P.), 482011, INDIA. E-mail:kushbohre@yaho.co.in

S. S. Thakur

Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, (M. P.), 482011, INDIA. E-mail:samajh_singh@rediffmail.com

Abstract: In this paper we introduce and characterize the concepts of upper and lower contra-continuous intuitionistic fuzzy multifunctions from a topological space to an intuitionistic fuzzy topological space.

Key words: Intuitionistic fuzzy sets, Intuitionistic fuzzy topology, Intuitionistic fuzzy multifunctions, lower contra-continuous and upper contra-continuous Intuitionistic fuzzy multifunctions.

Fuzzy Upper and Lower Almost Contra-continuous Multifunctions

Anjana Bhattacharyya

Department of Mathematics Victoria Institution (College) 78 B, A. P. C. Road Kolkata-700009, India. E-mail:anjanabhattacharyya@hotmail.com E-mail:amrishhanda83@gmail.com

Abstract: In this paper, a new type of fuzzy multifunction, viz. fuzzy upper (lower) almost contra-continuous multifunctions have been introduced and studied. Several charac-terizations of these newly defined fuzzy multifunctions have been made and shown that it is independent of fuzzy upper (lower) semi-continuous multifunctions [9].

Key words: Fuzzy upper (lower) almost contra-continuous multifunction, fuzzy semiopen set, fuzzy regular closed set.

A New Operator in Fuzzy Topological Spaces, Fuzzy Star-compact Spaces and Fuzzy Star-Lindel?f Spaces

Prasenjit Bal and Subrata Bhowmik

Department of Mathematics, Tripura University Suryamaninagar, Tripura, INDIA-799022. E-mail:balprasenjit177@gmail.com, subrata_bhowmik_math@rediffmail.com

Abstract: The concept of fuzzy sets was introduced in [9], that provides a framework for generalizing many of the concepts in general set theory, like general set theoretic topology which is called fuzzy topology introduced in [3]. Many researchers have studied the concept of compactness and Lindel?fness and their interesting properties like closedness, openness [1, 2, 3, 4, 8] etc in fuzzy topological spaces. The notion of star operator was introduced by E. K. van Douwen in [5]. Star-Lindel?f space was introduced by Song in [7]. We will generalize this spaces in the concept of fuzzy topological spaces.

Key words: Star-Lindel?f space, Fuzzy topological space, Fuzzy compact space, Fuzzy Lindel?f space.

Fuzzy Soft ¦Á-separation Axioms via Fuzzy ¦Á-open Soft Sets

A. M. Abd El-latif

Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt. E-mail:Alaa_8560@yahoo.com

Abstract: In the present paper, we continue the study on fuzzy soft topological spaces and investigate the properties of fuzzy¦Á-open (closed) soft sets, fuzzy ¦Á-soft interior (closure), fuzzy ¦Á-continuous (open) soft functions and fuzzy ¦Á-separation axioms which are important for further research on fuzzy soft topology. In particular we study the relationship between fuzzy ¦Á-soft interior fuzzy ¦Á-soft closure, which are basic for further research on fuzzy soft topology and will fortify the footing of the theory of fuzzy soft topological space.

Key words: Soft set, Fuzzy soft set, Fuzzy soft topological space, Fuzzy ¦Á-soft interior, Fuzzy ¦Á-soft closure, Fuzzy ¦Á-open soft, Fuzzy ¦Á-closed soft, Fuzzy ¦Á-continuous soft functions, Fuzzy soft ¦Á-separation axioms, Fuzzy soft ¦Á-T-spaces ( ), Fuzzy soft ¦Á-regular, Fuzzy soft ¦Á-normal.

Nearly C-compactness in Intuitionistic Fuzzy Topological Spaces

Mahima Thakur

Department of Applied Mathematics Jabalpur Engineering Colleger, Jabalpur, (M. P.) 482011 India. E-mail: mahimavthakur@gmail.com

S. S. Thakur

Department of Applied Mathematics Jabalpur Engineering College, Jabalpur, (M. P.) 482011 India. E-mail: samajh_singh@rediffimail.com

Abstract: In the present paper we extend the concept of fuzzy nearly C-compactness due to Chetty and Balasubramanian [9] in intuitionistic fuzzy topological spaces and obtain some of their characterizations and properties.

Key words: Intuitionistic fuzzy nearly C-compactness, Intuitionistic fuzzy filters.

S-connectedness in Intuitionistic Fuzzy Topological Spaces

S.S. Thakur and Mahima Thakur

Department of Applied Mathematics Jabalpur Engineering College, Jabalpur, (M. P.) 482011 India. E-mail: samajh_singh@rediffmail.com mahimavthakur@gmail.com

Shailendra Singh Thakur

In the present paper we introduce and discuss the concepts of intuitionistic fuzzy s-connectedness and intuitionistic fuzzy s-connectedness between intuitionistic fuzzy sets in intuitionistic fuzzy topological spaces. E-mail: ssthakur@ggits.org

Abstract: In this paper we investigate the solutions of linear time-varying differential dynamical systems with fuzzy initial condition and fuzzy inputs. We use a complex number representation of the ¦Á-level sets of the fuzzy states to characterize the solutions of such systems by a closed form formula which could be easily used in practical computations. Examples are given to illustrate the results.

Fuzzy Anti 2-inner Product Space and Its Properties

Parijat Sinha

Department of Mathematics, V. S. S. D. College, Kanpur

Divya Mishra and Ghanshyam Lal

Department of Mathematics, M. G. C. G. University, Satna India. E-mail:dvmishra275@gmail.com

Abstract: The purpose of this paper is to introduce the concept of fuzzy anti 2-inner product function on a linear space and its properties. The relative fuzzy anti 2-norm with respect to fuzzy anti 2-inner product has been defined. Parallelogram law and polarization identity has also been proved.

Key words: Fuzzy anti 2-inner product space, ¦Á-anti 2- norm, Parallelogram law.

On Fuzzy Soft Irresolute Functions

S. E. Abbas

Department of Mathematics, Faculty of Science, Jazan University, Saudi-Arabia.

E. El-sanowsy

Department of Mathematics, Faculty of Science, Sohag 82524, Egypt.

A. Atef

Preparatory Year Deanship, King Saud University, Saudi-Arabia.

Abstract: In this paper, semi-open and semi-closed sets on a fuzzy soft topological space in Sostak sense are introduced and some properties have been studied. Fuzzy soft semi-closure and fuzzy soft semi-interior operators are introduced and studied. Also, semi-continuous, semi-open, semi-closed and irresolute fuzzy soft functions on a fuzzy soft topological space are introduced and characterized.

Key words: r-fuzzy soft semi-open set, r-fuzzy soft semi-closed set, fuzzy soft semi-continuity, fuzzy soft irresolute mapping.

Boolean Algebraic Intuitionistic Fuzzy Topological Spaces

P. K. Sharma

Post Graduate Department of Mathematics, D.A.V. College, Jalandhar city, Punjab, India.E-mail:pksharma@davjalandhar.com

Abstract: The notion of intuitionistic fuzzy set was introduced by K.T. Atanassov as a generalization of the notion of fuzzy set. Intuitionistic fuzzy topological spaces were introduced by D. Coker and studied by many eminent authors like F. Gallego Lupianez, K. Hur, J. H. Kim and J. H. Ryou. R. Biswas applied the notion of intuitionistic fuzzy set to algebra and introduced intuitionistic fuzzy subgroup of a group. In this paper, we will study intuitionistic fuzzy topology by involving the Boolean algebraic structure on it and introduce the notion of Boolean algebraic intuitionistic fuzzy topological spaces. We will examine many properties of these spaces and obtain several results.

Key words: Intuitionistic fuzzy topological space (IFTS), intuitionistic fuzzy Boolean subalgebra (IFBSA), Boolean algebraic intuitionistic fuzzy topological space (BAIFTS), intuitionistic fuzzy point (IFP).