Matematik Bölümü / Department of Mathematics http://hdl.handle.net/20.500.11851/261 2020-08-04T03:38:12Z Some Results On Hausdorff Metric For Intuitionistic Fuzzy Sets http://hdl.handle.net/20.500.11851/3566 Some Results On Hausdorff Metric For Intuitionistic Fuzzy Sets Akın, Ömer; Bayeğ, Selami Fuzzy set theory was frstly introduced by L. A. Zadeh in 1965 . In fuzzy sets, every element in the set is accompanied with a function μ(x): X → [0, 1], called membership function. The membership function may have uncertainty in some applications because of the subjectivity of the expert or the missing information in the model. Hence some extensions of fuzzy set theory were proposed [2-4]. One of these extensions is Atanassov’s intuitionistic fuzzy set (IFS) theory . In 1986, Atanassov  introduced the concept of intuitionistic fuzzy sets and carried out rigorous researches to develop the theory. In this set concept, he introduced a new degree ν(x) : X → [0, 1], called non-membership function, such that the sum μ+ν is less than or equal to 1. Hence the difference 1 −(μ+ν) is regarded as degree of hesitation. Since intuitionistic fuzzy set theory contains membership function, non-membership function and the degree of hesitation, it can be regarded as a tool which is more flexible and closer to human reasoning in handling uncertainty due to imprecise knowledge or data. The Hausdorff metric distance between the alpha cuts of fuzzy numbers is one of the most used metric on fuzzy set theory. It measures how far fuzzy numbers are . In this work we will give some results based on the maximum metric involving Hausdorff metric to measure the distance between intuitionistic fuzzy numbers. We will show that intuitionistic fuzzy numbers are complete under the maximum metric based on Hausdorff metric. As a result we will prove that that space of continuous intuitionistic fuzzy number valued functions are complete under this metric. International Conference On Mathematics And Mathematics Educatİon ICME-2018, (27-28 June 2018: Ordu,Turkey) 2018-01-01T00:00:00Z An Indicator Operator Algorithm for Solving a Second Order Intuitionistic Fuzzy Initial Value Problem http://hdl.handle.net/20.500.11851/3565 An Indicator Operator Algorithm for Solving a Second Order Intuitionistic Fuzzy Initial Value Problem Akın, Ömer; Bayeğ, Selami L. Zadeh  was the first who introduced the concept of fuzzy settheory as an extension of the classical notion of the set theory. He remindedpeople that things are not always black or white; there may besome grey colours in life. Hence he simply assigned so called the membershipfunction to each element in a classical set and started the fuzzyset theory. However in some cases the membership concept in a fuzzyset is itself uncertain. This uncertainty may be because of the subjectivityof expert knowledge, complexity of data or imprecision of themodels. Since the fuzzy set theory considers only the degree of membership,it does not involve the degree of uncertainty for the membership.To handle such situations, the generalized concepts of fuzzy set theory are used -. One of these generalizations is intuitionistic fuzzy settheory which was given by Atanassov . Atanassov introduced intuitionisticfuzzy set concept by extending the definition of fuzzy set afteranalyzing the shortcomings of it. He defined the intuitionistic fuzzy setconcept by introducing the the nonmembership function into the fuzzyset such that sum of both is less than one. And in his further researches,he showed the exclusive properties of intuitionistic fuzzy sets -.In the past decades, the intuitionistic fuzzy set theory has penetratedinto different research areas, such as decision making -, clusteringanalysis , medical diagnosis -, pattern recognition - In recent years the topic of fuzzy differential equations has beenrapidly grown to model the real life situations where the observed datais insufficient -. Especially to describe the relation between velocityand acceleration, second order differential equations has greaterimportance in science. Therefore many approaches - were givento solve second order fuzzy differential equations. However there areonly few works - to observe the intuitionistic fuzzy differentialequations.In this work, we have examined the solution of the following second orderintuitionistic fuzzy initial value problems given in Eq. (1)-(2) usingintuitionistic Zadeh’s Extension Principle . 𝑦 ′′(𝑥) + 𝑎1 𝑖 𝑦 ′ (𝑥) + 𝑎2 𝑖 𝑦(𝑥) = ∑𝑏 ̅𝑗 𝑖𝑔𝑗(𝑥) 𝑟 𝑗=1 𝑦(0) = 𝛾 ̅0 𝑖 ; 𝑦′(0) = 𝛾 ̅1 𝑖 Here 𝛾 ̅0 𝑖 ; 𝛾 ̅1 𝑖 and 𝑏̅ 𝑗 𝑖 (j=1, 2, 3,…, r ) are intuitionistic fuzzy numbers. 𝑔𝑗(𝑥) (j=1, 2, 3,…,r ) are continuous forcing functions on the interval [0, ∞). International Conference On Mathematics And Mathematics Educatİon (ICMME-2016) (12-14 May 2016: Elazığ, Turkey) 2016-01-01T00:00:00Z Some Results on the Fundamental Concepts of Fuzzy Set Theory in Intuitionistic Fuzzy Environment by Using α and β cuts http://hdl.handle.net/20.500.11851/3564 Some Results on the Fundamental Concepts of Fuzzy Set Theory in Intuitionistic Fuzzy Environment by Using α and β cuts Akın, Ömer; Bayeğ, Selami In this paper we have frstly examined the properties of α and β cuts of intuitionistic fuzzy numbers in Rn with the help of well-known Stacking and Characterization theorems in fuzzy set theory. Then, we have studied the generalized Hukuhara diﬀerence in intuitionistic fuzzy environment by using the properties of α and β cuts and support function. Finally, we have extended the strongly generalized diﬀerentiability concept from fuzzy set theory to intuitionistic fuzzy environment and proved the related theorems with this concept. 2019-01-01T00:00:00Z Approximation by max-min operators: A general theory and its applications http://hdl.handle.net/20.500.11851/3563 Approximation by max-min operators: A general theory and its applications Gökçer, Yeliz Türkan; Duman, Oktay In this study, we obtain a general approximation theorem for max-min operators including many significant applications. We also study the error estimation in this approximation by using Hölder continuous functions. The main motivation for this work is the paper by Bede et al. (2008) . As a special case of our results, we explain how to approximate nonnegative continuous functions of one and two variables by means of the max-min Shepard operators. We also study the approximation by the max-min Bernstein operators. Furthermore, to verify the theory we display graphical illustrations. 2020-09-01T00:00:00Z