A. Karthika and I. Arockiarani / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.2603-2607

Fuzzy C -Semi-Boundary in IFT Spaces A. Karthika and I. Arockiarani Nirmala College for women Coimbatore India

Abstract In this paper we introduce a new concept, Intuitionistic fuzzy đ?&#x2022;° Semi -boundary using the arbitrary complement function namely đ?&#x2022;°: [0,1]â&#x2020;&#x2019; [0,1] which is introduced by klir[10]. Further we investigate some of the properties of this boundary.

Keywords and phrases: Intuitionistic fuzzy set, Intuitionistic fuzzy topological spaces, boundary, Fuzzy semi-pre boundary.

Fuzzy

Mathematics Subject Classification: 54A40

I.

Introduction

Athar and Ahmad[2] defined the notion of fuzzy boundary in FTS and studied the properties of fuzzy semi boundary. K.Bageerathi and T.Thangavelu were introduced fuzzy đ?&#x201D;&#x2C6;-semi closed sets in fuzzy topological spaces, where đ?&#x201D;&#x2C6; â&#x2C6;ś 0,1 â&#x2020;&#x2019;[0,1] is an arbitrary complement function which satisfies the monotonic and involutive condition. After the introduction of fuzzy sets by Zadeh [13], there has been a number of generalizations of this fundamental concept. The notion of intuitionistic fuzzy sets was introduced by Atanassov [1] Using this idea Coker[9] introduced the concept of intuitionistic fuzzy topological spaces. For the past few years many concepts in fuzzy topology were extended to IFTS. In this paper, we are extending the concept of intuitionistic fuzzy semi boundary by using the complement function đ?&#x201D;&#x2C6; instead of the usual complement function. Further we discuss about the basic properties and some of the characterizations of this intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi boundary. II. Preliminaries Throughout this paper (X, đ?&#x153;?) denotes a intuitionistic topological space in which no separation axioms are assumed unless otherwise mentioned. Definition 2.1[1] Let X be a non-empty fixed set. An intuitionistic fuzzy set (IFS, for short), A is an object having the form A = {<x, đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ , đ?&#x203A;žđ??´ (đ?&#x2018;Ľ)> : xâ&#x2C6;&#x2C6; X} Where the mapping đ?&#x153;&#x2021;đ??´ â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; đ??ź and đ?&#x203A;žđ??´ â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; đ??ź denotes respectively the degree of membership (namely đ?&#x153;&#x2021;đ??´ (đ?&#x2018;Ľ)) and the non-membership

(namelyđ?&#x203A;žđ??´ (đ?&#x2018;Ľ) ) of each element x â&#x2C6;&#x2C6;X to a set A, and 0 â&#x2030;¤ đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ + đ?&#x203A;žđ??´ đ?&#x2018;Ľ â&#x2030;¤ 1 for each xâ&#x2C6;&#x2C6;X. A = {<x, đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ , đ?&#x203A;žđ??´ đ?&#x2018;Ľ > : x â&#x2C6;&#x2C6; X Definition 2.2[1] Let X be a non-empty set and let the IFSâ&#x20AC;&#x2122;s A and B in the form A = A = {<x, đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ , đ?&#x203A;žđ??´ đ?&#x2018;Ľ > : x â&#x2C6;&#x2C6; X} ; B = A = {<x, đ?&#x153;&#x2021;đ??ľ đ?&#x2018;Ľ , đ?&#x203A;žđ??ľ đ?&#x2018;Ľ > : x â&#x2C6;&#x2C6; X}. Let {đ??´đ?&#x2018;&#x2014; â&#x2C6;ś đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ??˝} be an arbitray family of IFSâ&#x20AC;&#x2122;s in (X, đ?&#x153;?). Then , (i) đ??´ â&#x2030;¤ đ??ľ if and only if â&#x2C6;&#x20AC; đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039; , đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ â&#x2030;¤ đ?&#x153;&#x2021;đ??ľ (đ?&#x2018;Ľ) and đ?&#x203A;žđ??´ (đ?&#x2018;Ľ) â&#x2030;Ľ đ?&#x203A;žđ??ľ (đ?&#x2018;Ľ) (ii) đ??´ = {< đ?&#x2018;Ľ, đ?&#x203A;žđ??´ đ?&#x2018;Ľ , đ?&#x153;&#x2021;đ??´ (đ?&#x2018;Ľ) >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} (iii) â&#x2C6;Š đ??´đ?&#x2018;&#x2014; = {< đ?&#x2018;Ľ, â&#x2C6;§ đ?&#x153;&#x2021;đ??´đ?&#x2018;&#x2014; (đ?&#x2018;Ľ) ,â&#x2C6;¨ đ?&#x203A;žđ??´đ?&#x2018;&#x2014; (đ?&#x2018;Ľ) >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} (iv) â&#x2C6;Ş đ??´đ?&#x2018;&#x2014; = { < đ?&#x2018;Ľ, â&#x2C6;¨ đ?&#x153;&#x2021;đ??´đ?&#x2018;&#x2014; (đ?&#x2018;Ľ) , â&#x2C6;§ đ?&#x203A;žđ??´đ?&#x2018;&#x2014; (đ?&#x2018;Ľ) >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} >} (v) 1 = { < đ?&#x2018;Ľ, 1, 0 >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} and 0 = {< đ?&#x2018;Ľ, 0, 1 >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;} (vi) đ??´ = đ??´, 1 = 0 and 0 = 1. Definition 2.3[9] Let X and Y be two non-empty sets and đ?&#x2018;&#x201C;: X, đ?&#x153;? â&#x2020;&#x2019; (X, đ?&#x153;&#x17D;) be a mapping. If đ??ľ = {< y, đ?&#x153;&#x2021;đ??ľ đ?&#x2018;Ś , đ?&#x203A;žđ??ľ đ?&#x2018;Ś > : y â&#x2C6;&#x2C6; Y} is an IFS in Y, then the preimage of B under f is denoted and defined by đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 đ??ľ = {< đ?&#x2018;Ľ, đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x153;&#x2021;đ??ľ đ?&#x2018;Ľ ), đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ?&#x203A;žđ??ľ đ?&#x2018;Ľ ) >â&#x2C6;ś đ?&#x2018;Ľ â&#x2C6;&#x2C6; X } since đ?&#x153;&#x2021;đ??ľ , đ?&#x203A;žđ??ľ are fuzzy sets, we explain that đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 đ?&#x153;&#x2021;đ??ľ đ?&#x2018;Ľ = đ?&#x153;&#x2021;đ??ľ (đ?&#x2018;&#x201C; đ?&#x2018;Ľ ) Definition 2.4[9] An instuitionistic fuzzy topology (IFT, for short) on a non-empty set X is a family đ?&#x153;? of IFSâ&#x20AC;&#x2122;s in X satisfying the following axioms: (i) 1, 0 â&#x2C6;&#x2C6; đ?&#x153;? (ii) đ??´1 â&#x2C6;Š đ??´2 â&#x2C6;&#x2C6; đ?&#x153;? for some đ??´1 , đ??´2 â&#x2C6;&#x2C6; đ?&#x153;? (iii) â&#x2C6;Ş đ??´đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ?&#x153;? for any {đ??´đ?&#x2018;&#x2014; : đ?&#x2018;&#x2014; â&#x2C6;&#x2C6; đ??˝} â&#x2C6;&#x2C6; đ?&#x153;? In this case, the ordered pair (X, đ?&#x153;?) is called intuitionistic fuzzy topological space(IFTS, for short) and each IFS in đ?&#x153;? is known as intuitionistic fuzzy open set(IFOS, for short) in X. The complement of an intuitionistic fuzzy open set is called intuitionistic fuzzy closed set(IFCS, for short). Definition 2.5[9] Let (X, đ?&#x153;?) be an IFTS and let A = {<x, đ?&#x153;&#x2021;đ??´ đ?&#x2018;Ľ , đ?&#x203A;žđ??´ đ?&#x2018;Ľ > : x â&#x2C6;&#x2C6; X} be oa IFS in X. Then intuitionistic fuzzy interior and intuitionistic fuzzy closure of A are defined by đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ??´ = â&#x2C6;Ş {đ??ş â&#x2C6;ś đ??ş đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ??źđ??šđ?&#x2018;&#x201A;đ?&#x2018;&#x2020; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; đ?&#x2018;&#x2039; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??ş â&#x160;&#x2020; đ??´}

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A. Karthika and I. Arockiarani / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.2603-2607 đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ = â&#x2C6;Š {đ??ž â&#x2C6;ś đ??ž đ?&#x2018;&#x2013;đ?&#x2018; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A; đ??źđ??šđ??śđ?&#x2018;&#x2020; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A; đ?&#x2018;&#x2039; đ?&#x2018;&#x17D;đ?&#x2018;&#x203A;đ?&#x2018;&#x2018; đ??´ â&#x160;&#x2020; đ??ž } Remark 2.6 For any IFS A in (X, đ?&#x153;?), we have đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ = đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ??´ and đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ??´ = đ?&#x2018;?đ?&#x2018;&#x2122; đ??´. Definition 2.7[5] Let đ?&#x201D;&#x2C6; â&#x2C6;ś 0,1 â&#x2020;&#x2019; [0,1] be a complement function. If A is a fuzzy subset of (X, đ?&#x153;?) then the complement đ?&#x201D;&#x2C6;đ??´ of a fuzzy subset A is defined by đ?&#x201D;&#x2C6;đ??´ đ?&#x2018;Ľ = đ?&#x201D;&#x2C6; A x for all đ?&#x2018;Ľ â&#x2C6;&#x2C6; đ?&#x2018;&#x2039;. A complement function đ?&#x201D;&#x2C6; is said to satisfy (i) The boundary condition if đ?&#x201D;&#x2C6; 0 = 1 and đ?&#x201D;&#x2C6; 1 = 0, (ii) Monotonic condition if đ?&#x2018;Ľ â&#x2030;¤ đ?&#x2018;Ś â&#x2021;&#x2019; đ?&#x201D;&#x2C6; đ?&#x2018;Ľ â&#x2030;Ľ đ?&#x201D;&#x2C6; (y) for all x, y â&#x2C6;&#x2C6;[0,1] (iii) Involutive condition if đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6; x = đ?&#x2018;Ľ for all x â&#x2C6;&#x2C6; [0,1]. Lemma 2.8[5] Let đ?&#x201D;&#x2C6;: 0,1 â&#x2020;&#x2019; [0,1] be a complement function that satisfies the monotonic and involutive condition. Then for any family {đ?&#x153;&#x2020;đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; Î&#x201D;} of fuzzy subsets of X, we have (i) đ?&#x201D;&#x2C6; sup đ?&#x153;&#x2020;đ?&#x203A;ź đ?&#x2018;Ľ : đ?&#x203A;ź â&#x2C6;&#x2C6; Î&#x201D; = inf đ?&#x201D;&#x2C6; ÎťÎą x : đ?&#x203A;ź â&#x2C6;&#x2C6; Î&#x201D;=inf{ đ?&#x201D;&#x2C6; ÎťÎąđ?&#x2018;Ľ: đ?&#x203A;źâ&#x2C6;&#x2C6; Î&#x201D;} and (ii) đ?&#x201D;&#x2C6; inf đ?&#x153;&#x2020;đ?&#x203A;ź đ?&#x2018;Ľ : đ?&#x203A;ź â&#x2C6;&#x2C6; Î&#x201D; = sup đ?&#x201D;&#x2C6; ÎťÎą x : đ?&#x203A;ź â&#x2C6;&#x2C6; Î&#x201D;=sup{ đ?&#x201D;&#x2C6; ÎťÎąđ?&#x2018;Ľ: đ?&#x203A;źâ&#x2C6;&#x2C6; Î&#x201D;} for x â&#x2C6;&#x2C6; X. Definition 2.9[5] A fuzzy subset A of X is fuzzy đ?&#x201D;&#x2C6;-closed in (X, đ?&#x153;?) if đ?&#x201D;&#x2C6;A is fuzzy open in (X, đ?&#x153;?). The fuzzy đ?&#x201D;&#x2C6;closure of A is defined as the intersction of all fuzzy đ?&#x201D;&#x2C6;-closed sets B containing A. The fuzzy đ?&#x201D;&#x2C6;-closure of A is denoted by đ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (đ??´) that is equal to â&#x2C6;§ {đ??ľ: đ??ľ â&#x2030;Ľ đ??´, đ?&#x201D;&#x2C6;đ??ľ â&#x2C6;&#x2C6; đ?&#x153;? }. Lemma 2.10[5] If the complement function đ?&#x201D;&#x2C6;satisfies the monotonic and involutive conditions, then for any fuzzy subset A of X, đ?&#x201D;&#x2C6; đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ??´ = đ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (đ?&#x201D;&#x2C6;đ??´) and đ?&#x201D;&#x2C6; đ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ = đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą ( đ?&#x201D;&#x2C6;đ??´). Lemma 2.11[5] Let (X, đ?&#x153;?) be a fuzzy topological space. Let đ?&#x201D;&#x2C6; be a complement function that satisfies that satisfies the boundary, monotonic and involutive conditions. Then for any family {đ??´đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; â&#x2C6;&#x2020;} of fuzzy subsets of X. We have đ?&#x201D;&#x2C6; â&#x2039; đ??´đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; â&#x2C6;&#x2020; = â&#x2C6;§ { đ?&#x201D;&#x2C6;đ??´đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; â&#x2C6;&#x2020;} and đ?&#x201D;&#x2C6; â&#x2C6;§ đ??´đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; â&#x2C6;&#x2020; = â&#x2C6;¨ { đ?&#x201D;&#x2C6;đ??´đ?&#x203A;ź : đ?&#x203A;ź â&#x2C6;&#x2C6; â&#x2C6;&#x2020;}. Definition 2.12[5] Let (X, đ?&#x153;?) be a fuzzy topological space and đ?&#x201D;&#x2C6; be a complement function. Then for a fuzzy subset A of X, the fuzzy đ?&#x201D;&#x2C6;-semi interior of A(briefly sđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x201D;&#x2C6; đ??´), is the union of all fuzzy đ?&#x201D;&#x2C6;-semi open sets of X contained in A.

That is open}.

sđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x201D;&#x2C6;

đ??´ = â&#x2039; { đ??´ â&#x2C6;ś đ??ľ â&#x2030;¤ đ??´ , B is fuzzy đ?&#x201D;&#x2C6;-semi

Definition 2.13[5] Let (X, đ?&#x153;?) be a fuzzy topological space and đ?&#x201D;&#x2C6; be a complement function. Then for a fuzzy subset A of X, the fuzzy đ?&#x201D;&#x2C6;-semi closure of A(briefly sđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??´), is the union of all fuzzy đ?&#x201D;&#x2C6;-semi open sets of X contained in A. That is sđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??´ = â&#x2C6;¨ { đ??ľ: đ??ľ â&#x2030;Ľ đ??´ , B is fuzzy đ?&#x201D;&#x2C6;-semi closed}. Lemma 2.14[2] Let (X, đ?&#x153;?) and (Y, đ?&#x153;&#x17D;) be đ?&#x201D;&#x2C6;-product related fuzzy topological spaces. Then A is called fuzzy đ?&#x201D;&#x2C6;semi open if there exists a đ?&#x201D;&#x2C6;-open set B such that đ??ľ â&#x2030;¤ đ??´ â&#x2030;¤ đ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ. Lemma 2.15[4] If đ??´1 , đ??´2 , đ??´3 , đ??´4 are the fuzzy subsets of X then đ??´1 â&#x2C6;§ đ??´2 đ?&#x2018;&#x2039; đ??´3 â&#x2C6;§ đ??´4 = đ??´1 đ?&#x2018;&#x2039; đ??´4 â&#x2C6;§ (đ??´2 đ?&#x2018;&#x2039; đ??´3 ). Lemma 2.16[4] Suppose f is a function from X to Y. Then đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 đ?&#x201D;&#x2C6;đ??ľ = đ?&#x201D;&#x2C6;(đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ??ľ)) for any fuzzy subset B of Y. Definition 2.17[12] If A is a fuzzy subset of X and B is a fuzzy subset of Y, then AxB is a fuzzy subset of XxY defined by (AxB)(x,y) = min {A(x), B(y)} for each (x,y) â&#x2C6;&#x2C6; XxY. Lemma 2.18[4] Let đ?&#x2018;&#x201C; â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; đ?&#x2018;&#x152; be a function. If {đ??´đ?&#x203A;ź } a family of fuzzy subsets of Y, then (i) đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 â&#x2C6;¨ đ??´đ?&#x203A;ź = â&#x2C6;¨ đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ??´đ?&#x203A;ź ) (ii) đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 â&#x2C6;§ đ??´đ?&#x203A;ź = â&#x2C6;§ đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ??´đ?&#x203A;ź ) Lemma 2.19[4] If A is an IFS of X and B is a IFS of Y, then đ?&#x201D;&#x2C6;(A x B) = đ?&#x201D;&#x2C6;A x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;B.

III.

Intuitionistic fuzzy đ?&#x2022;°-semi boundary

In this section intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closure and interior were introduced. Using these two concepts we develop the idea of intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi boundary. Definition 3.1 Let (X, đ?&#x153;?) be a IFTS. Let A be a IFS and let đ?&#x201D;&#x2C6; be the complement function. Then intuitionistic fuzzy đ?&#x201D;&#x2C6;semi boundary is defined as isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; ( đ?&#x201D;&#x2C6;đ??´), where semi closure and semi interior were defined as isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ąđ?&#x201D;&#x2C6; đ??´ = â&#x2039; { đ??´ â&#x2C6;ś đ??ľ â&#x2030;¤ đ??´ , B is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open}. isđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??´ = â&#x2C6;¨ { đ??ľ: đ??ľ â&#x2030;Ľ đ??´ , B is intuitionistic fuzzy đ?&#x201D;&#x2C6;semi closed}. Proposition 3.2

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A. Karthika and I. Arockiarani / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.2603-2607 Let (X, đ?&#x153;?) be a IFTS and đ?&#x201D;&#x2C6; b isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ e a complement function that satisfies the monotonic and involutive condition. Then for any IFS A of X, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ . Proof

By the definition 3.1, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ đ?&#x201D;&#x2C6;đ??´). Since đ?&#x201D;&#x2C6; satisfies the involutive condition đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ = đ??´, that implies isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (đ?&#x201D;&#x2C6;đ??´) â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;( đ?&#x201D;&#x2C6;đ??´). That is isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ . isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (

the monotonic and involutive conditions, we have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ . Proposition 3.7 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A of X, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ Î˛đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ . Proof

Let A be intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed, and đ?&#x201D;&#x2C6; satisfies the involutive and monotonic conditions we have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ isđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??´ = đ??´.

Since đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions, by using proposition 3.6, we have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ]= isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ [isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ] â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ]. Since isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semiopen, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ]= đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ]. Since isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ [isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ] â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ đ??´, by using proposition 5.6(ii) in [6], and proposition 5.5 in [6], isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [ đ?&#x201D;&#x2C6;đ??´]. By using the definition 3.1, we have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ .

Proposition 3.4 Let (X, đ?&#x153;?) be a IFTS and đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. If A is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open, then isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ đ?&#x201D;&#x2C6; đ??´.

Proposition 3.8 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A of X, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ .

Proof

Proof

Proposition 3.3 Let (X, đ?&#x153;?) be a IFTS and đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. If A is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed, then isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ đ??´. Proof

Let A be intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open. Since đ?&#x201D;&#x2C6; satisfies the involutive condition, we get, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ đ?&#x201D;&#x2C6;đ??´. Proposition 3.5 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function satisfies the monotonic and involutive conditions. Then for any IFS A of X, we have đ?&#x201D;&#x2C6; isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ â&#x2C6;¨ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ . Proof

Using Definition 3.1, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ đ?&#x201D;&#x2C6;đ??´) . Taking complement on both sides, we get đ?&#x201D;&#x2C6; [isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ] = đ?&#x201D;&#x2C6; [isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; ( đ?&#x201D;&#x2C6;đ??´)] . Since đ?&#x201D;&#x2C6; the monotonic and involutive conditions, đ?&#x201D;&#x2C6; [ Using isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ] = đ?&#x201D;&#x2C6;[isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´] â&#x2C6;¨ đ?&#x201D;&#x2C6;[ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ ]. proposition 5.6 in [6], đ?&#x201D;&#x2C6; isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ â&#x2C6;¨ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ . isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (

Proposition 3.6 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A of X, we have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ . Proof isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6;

By the Definition 3.1, đ??´ =isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ . Since

we have đ?&#x201D;&#x2C6; satisfies

Since đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions, by using proposition 3.11, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ]= đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ [ đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ ] â&#x2C6;§ đ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą [ đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ ]. By using is đ?&#x201D;&#x2C6; is đ?&#x201D;&#x2C6; is đ?&#x201D;&#x2C6; is đ?&#x201D;&#x2C6; proposition 5.6(iii) in [6], we have isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] = isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ , that implies isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] = isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ]. Since A â&#x2030;¤[ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] , that implies isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ â&#x2030;¤ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ]. Therefore isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; (đ??´). By using proposition 5.5(ii) in [6], and using the definition 3.1, we get isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ . Theorem 3.9 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A and B of X, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;¨ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??ľ . Proof From the definition 3.1, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ]= isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ] â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;[đ??´ â&#x2C6;¨ đ??ľ] . Since đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions, by the proposition 5.7(i) in [6], that implies isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ]= {isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;¨ [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]}â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;[đ??´ â&#x2C6;¨ đ??ľ] . By using , proposition 5.7 (ii) in [6], isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ] â&#x2030;¤ {isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;¨ [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]}â&#x2C6;§ { isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??ľ }. That is isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ] â&#x2030;¤ {isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;§ [ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??ľ }. Again by isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ ]}â&#x2C6;¨ { isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´

2605 | P a g e

A. Karthika and I. Arockiarani / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.2603-2607 using definition 3.1, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??ľ .

isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´

â&#x2C6;¨ đ??ľ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;¨

Theorem 3.10 Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A and B of X, {isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;§ [ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ đ??ľ] â&#x2030;¤ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ }. isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]}â&#x2C6;¨ { isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??ľ Proof

By 3.1 isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ đ??ľ]= isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ đ??ľ] â&#x2C6;§ đ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6;[đ??´ â&#x2C6;§ đ??ľ] . Since đ?&#x201D;&#x2C6; satisfies the monotonic is đ?&#x201D;&#x2C6; and involutive conditions, by using propositions 5.7(i), in [6] and by using lemma 2.11, we get {isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;§ [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]}â&#x2C6;§ { isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ đ??ľ] â&#x2030;¤ â&#x2C6;¨ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??ľ } is equal to isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ đ??ľ] â&#x2030;¤ {{isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;§ [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??´ ]}â&#x2C6;§ [isđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??´] } â&#x2C6;¨ {[ â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6;đ??ľ ] â&#x2C6;§ [isđ?&#x2018;?đ?&#x2018;&#x2122;đ?&#x201D;&#x2C6; đ??ľ]}. Again by isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ definition 3.1, we get {isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´] â&#x2C6;§ [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]}â&#x2C6;¨ { isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;§ đ??ľ] â&#x2030;¤ đ??ľđ?&#x2018;&#x2018; đ??ľ â&#x2C6;§ đ?&#x2018;?đ?&#x2018;&#x2122; đ??´ }. is is đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6; Proposition 3.11 Let Let (X, đ?&#x153;?) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A of X,we have 1. isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ . 2. [ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ]{isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ]} â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ . Proof

From the definition 3.1, isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´= isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ â&#x2C6;§ đ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6;đ??ľ. We have isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ = isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [ is đ?&#x201D;&#x2C6; isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´] â&#x2C6;§ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´ ] â&#x2030;¤ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [đ??´ â&#x2C6;¨ đ??ľ] isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; [isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´]. Since đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions, we have [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ]= A, where A is A is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed. Hence the proof. Proposition 3.12 Let A be a intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed subset of a IFTS X and B be a intuitionistic fuzzy đ?&#x201D;&#x2C6;semi closed subset od a IFTS Y, then A x B is a intuitionistic fuzzy đ?&#x201D;&#x2C6;-seml closed set of the intuitionistic fuzzy product space X x Y where the complement function đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions. Proof

Let A be a intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed subset of a IFTS X. Then đ?&#x201D;&#x2C6; đ??´ is intuitionistic fuzzy semi open in X. Then đ?&#x201D;&#x2C6;đ??´ x 1 is intuitionistic fuzzy semi open in X x Y. Similarly let B be a intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi closed subset od a IFTS Y, then 1 x đ?&#x201D;&#x2C6;đ??ľ is intuitionistic fuzzy semi open in X x Y. Since the arbitrary union of intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open sets is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open , đ?&#x201D;&#x2C6;đ??´ đ?&#x2018;Ľ 1 â&#x2C6;¨

1 đ?&#x2018;Ľ đ?&#x201D;&#x2C6;đ??ľ is intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open in X x Y. Then by lemma 2.19 đ?&#x201D;&#x2C6; (A x B) = đ?&#x201D;&#x2C6; A x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6; B, hence đ?&#x201D;&#x2C6; (A x B ) is intuitionistic fuzzy đ?&#x201D;&#x2C6; -semi open. Therefore A x B is intuitionistic fuzzy đ?&#x201D;&#x2C6; -semi closed of intuitionistic fuzzy product space X x Y. Theorem 3.13 Let (X, đ?&#x153;? ) be a IFTS. Let đ?&#x201D;&#x2C6; be a complement function that satisfies the monotonic and involutive conditions. Then for any IFS A and B of X,

(i) (ii)

isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6;

đ??´ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ â&#x2030;Ľ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) isđ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??ľ â&#x2030;¤ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; (A x B)

proof

By the definition 2.17,[ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A) x đ?&#x2018;?đ?&#x2018;&#x2122; đ??ľ ] (đ?&#x2018;Ľ, đ?&#x2018;Ś)= min { isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´(x), isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ(y)}â&#x2030;Ľ is đ?&#x201D;&#x2C6; min { A(x), B(y)} = (A x B ) (x,y). This shows that isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ â&#x2030;Ľ (A x B). By using proposition 5.6 in [6], isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) â&#x2030;¤ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ )= isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; ( isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ . By using definition 2.17 ,[ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; (A) x min { isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??ľ ] (đ?&#x2018;Ľ, đ?&#x2018;Ś)= isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´(x), isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??ľ(y)} â&#x2030;¤ min { A(x), B(y)} = (A x B ) (x,y). This shows that isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??ľ â&#x2030;¤ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; (A x B). Theorem 3.14 Let X and Y be đ?&#x201D;&#x2C6;-product related IFTSs. Then for a IFS A of X and a IFS B of Y, (i) isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) = isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ (ii) x isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; (A x B) = isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??´ isđ?&#x2018;&#x2013;đ?&#x2018;&#x203A;đ?&#x2018;Ą đ?&#x201D;&#x2C6; đ??ľ . Proof By the theorem 3.13, it is sufficient to show that isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ â&#x2030;¤ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6;(A x B). By using the definition in [6], we have isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) = inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź x đ??ľđ?&#x203A;˝ ) â&#x2C6;ś đ?&#x201D;&#x2C6;( A x B) â&#x2030;Ľ A x B where đ??´đ?&#x203A;ź and đ??ľđ?&#x203A;˝ are intuitionistic fuzzy đ?&#x201D;&#x2C6;-semi open }. By using lemma 2.17, we have, isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) = inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) â&#x2C6;ś đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) â&#x2030;Ľ A x B} =inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) : đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) â&#x2030;Ľ A or đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ â&#x2030;ĽB} = min ( inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) : đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) â&#x2030;Ľ A }, inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) : đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ â&#x2030;ĽB}). Now inf{ đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x đ?&#x201D;&#x2C6;(đ??ľđ?&#x203A;˝ ) : đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) â&#x2030;Ľ A} â&#x2030;Ľ inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 : đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) â&#x2030;Ľ A} = inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) : đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) â&#x2030;Ľ A} x 1 = [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´ ] x 1. 2606 | P a g e

A. Karthika and I. Arockiarani / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 4, Jul-Aug 2013, pp.2603-2607 Also inf { đ?&#x201D;&#x2C6;( đ??´đ?&#x203A;ź ) x 1 â&#x2C6;¨ 1 x (đ??ľđ?&#x203A;˝ ) : đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ â&#x2030;ĽB} â&#x2030;Ľ inf { 1 x đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ : đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ â&#x2030;ĽB} = 1 x inf { đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ : đ?&#x201D;&#x2C6; đ??ľđ?&#x203A;˝ â&#x2030;ĽB} = 1 x [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ ]. Thus we conclude the result as , isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (A x B) â&#x2030;Ľ min ([ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´] x 1 , 1 x [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ] = [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´]x [ isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??ľ]. Theorem 3.15 Let đ?&#x2018;&#x2039;đ?&#x2018;&#x2013; , i= 1, 2, 3, â&#x20AC;Ś.. n be a family of đ?&#x201D;&#x2C6;-product related IFTSs. If đ??´đ?&#x2018;&#x2013; is a IFS of đ?&#x2018;&#x2039;đ?&#x2018;&#x2013; , and the complement function đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions, then đ?&#x2018;&#x203A; x isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; [ đ?&#x2018;&#x2013;=1 đ??´đ?&#x2018;&#x2013; ]={ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; (đ??´1 ) x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´2 â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś x { isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (đ??´1 ) x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´đ?&#x2018;&#x203A; } â&#x2C6;¨ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; đ??´2 x â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´đ?&#x2018;&#x203A; } â&#x2C6;¨ â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś. â&#x2C6;¨ { isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; (đ??´1 ) x x â&#x20AC;Śâ&#x20AC;Śâ&#x20AC;Ś x isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ??´2 đ??ľđ?&#x2018;&#x2018; đ??´ }. Î˛is đ?&#x201D;&#x2C6; đ?&#x2018;&#x203A; Proof Using lemma 2.14, lemma 2.19, we get the result. Theorem 3.16 Let đ?&#x2018;&#x201C; â&#x2C6;ś đ?&#x2018;&#x2039; â&#x2020;&#x2019; đ?&#x2018;&#x152; be a intutionistic fuzzy continuous function. Suppose the complement function đ?&#x201D;&#x2C6; satisfies the monotonic and involutive conditions, then â&#x2C6;&#x2019;1 đ??ľ â&#x2030;¤ đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6;[B]), for any IFS B isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; (đ?&#x2018;&#x201C; in Y.

continuity, J. Math. Anal .Appl. 82(1) (1981), 14-32. [5] K.Bageerathi, G. Sutha, P. Thangavelu, A generalization of fuzzy closed sets, International Journal of fuzzy systems and rough systems, 4(1) (2011), 1-5. [6] K. Bageerathi, P. Thangavelu, A generalization of fuzzy Boundary, International Journal of Mathematical Archive â&#x20AC;&#x201C; 1(3)(2010), 73-80. [7] K.Bageerathi, On fuzzy semi-pre-boundary, International journal of Mathematical Archieve, 3(12)(2012), 4848-4855. [8] C.L. Chang, Fuzzy topological spaces, Journal of mathematical Analysis and Applications, vol. 24(1)(1968), 182-190. [9] D.Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy sets and systems, 88, (1997), 81-89. [10] George J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice â&#x20AC;&#x201C; Hall, Inc, 2005. [11] H.Gurcay and D.Coker, On fuzzy continuity in intuitionistic fuzzy topological spaces, J.Fuzzy Math 2(5)(1999), 375-384. [12] K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J.Math.Anal.Appl. 8 (3) (1978), 459-470 [13] L.A.Zadeh, Fuzzy sets, Information and control, (8)(1965), 338-353.

Proof Let f be a IF continuous function and B be a IFS in Y. By using definition 3.1, we have â&#x2C6;&#x2019;1 â&#x2C6;&#x2019;1 (đ??ľ)) = (đ??ľ ) â&#x2C6;§ [ isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; (đ?&#x2018;&#x201C; isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (đ??ľ )). Then using the lemmaâ&#x20AC;&#x2122;s isđ?&#x2018;?đ?&#x2018;&#x2122; đ?&#x201D;&#x2C6; đ?&#x201D;&#x2C6; (đ?&#x2018;&#x201C; 2.16, 2.18, finally we get isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6; (đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 đ??ľ â&#x2030;¤ đ?&#x2018;&#x201C; â&#x2C6;&#x2019;1 (isđ??ľđ?&#x2018;&#x2018; đ?&#x201D;&#x2C6;[B]). Hence the result. References [1] [2]

[3]

[4]

K.Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20,(1998), 87-96. M. Athar and B. Ahmad, Fuzzy boundary and Fuzzy semi boundary, Advances in Fuzzy systems(2008), Article ID586893, 9 pages. M. Athar and B. Ahmad, Fuzzy sets, fuzzy S-open and fuzzy S-closed mappings, Advances in Fuzzy systems(2009), Article ID 303042, 5 pages. K.K. Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly

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