On Irresolute Intuitionistic Fuzzy Multifunctions

Int J Contemp Math Sciences, Vol 7, 2012, no 21, 1013-1028 On Irresolute Intuitionistic Fuzzy Multifunctions S S Thakur Department of Applied Mathematics Government Engineering College Jabalpur (MP) 482011 India samajh_singh@rediffmailcom Kush Bohre Department of Applied Mathematics Gyan Ganga College of Technology Jabalpur (MP) - 482001 India kushbohre@yahoocoin Abstract The aim of this paper is introduce the concepts of upper and lower intuitionisic fuzzy irresolute intuitionistic fuzzy multi functions and obtain some of their properties Mathematics Subject Classification: 54A99, 03E99 Keywords: Intuitionistic fuzzy sets, Intuitionistic fuzzy topology, Intuitionistic fuzzy multifunctions, lower Intuitionistic fuzzy irresolute and upper Intuitionistic fuzzy irresolute Intuitionistic fuzzy multifunctions 1 Introduction After the introduction of fuzzy sets by Zadeh [31] in 1965 and fuzzy topology by Chang [7] in 1967, several researches were conducted on the generalizations of the notions of fuzzy sets and fuzzy topology The concept of intuitionistic fuzzy sets was introduced by Atanassov [1,2,3] as a generalization of fuzzy sets In the last 27 years various concepts of fuzzy mathematics have been extended for intuitionistic fuzzy sets In 1997 Coker [8] introduced the concept of intuitionistic fuzzy topological spaces as a generalization of fuzzy topological spaces Recently many fuzzy topological concepts such as fuzzy compactness [10], fuzzy connectedness [30], fuzzy separation axioms [4,16,18], fuzzy nets and filters [17], fuzzy metric spaces [29], and fuzzy continuity [13] have been generalized for intuitionistic fuzzy topological spaces In 1999, Ozbakir and Coker [24] introduced the concept intuitionistic fuzzy multifunctions and studied their lower and upper intuitionistic fuzzy semi continuity from a topological space

1014 S S Thakur and K Bohre to an intuitionistic fuzzy topological spacein the present paper we extend theconcepts of lower and upper irresolute multifunctions due to Popa [25] to intuitionistic fuzzy multifunctions obtain some of their characterizations and properties 2 Preliminaries Throughout this paper and (Y, ) represents a topological space and an intuitionistic fuzzy topological space respectively Definition 21[ 15 ] : A subset A of a topological space ) is called : (a) Semi open if there exists an open set U of X such that U A Cl(U ) (b) Semi closed if its complement A c is semi open The family of all semi open (resp semi closed) subsets of topological space is denoted by SO(X) (resp SC(X)) The intersection of all semiclosed sets of X containing a set A of X is called the semi closure [ 6] of A It is denoted by scl(a) The union of all semi open subsets of A of X is called the semi interior of A It is denoted by sint(a) A subset A of X is semi open (resp semi closed) if and only if A A subset N of a topological space ) is called a semi neighbourhood [6 ] of a point x of X if there exists a semi open set O of X such that x N A is a semi open in X if and only if it is a semi neighbourhood of each of its points Let A be a sub set of a topological space X A sub set V of X is called a semi-neighbourhood of a subset A of X if there exists SO(X) such that A mapping f from a topological space (X, to another topological space (X*, is said to be irresolute [11 ] (resp semi continuous [15]) if the inverse image of every semi open (resp open) set of X* is semi open in X Every continuous (resp irresolute ) mapping is semi continuous but the converse may not be true [ 11 ] The concepts of continuous and irresolute mappings are independent [11] Definition 21 [1, 2, 3 ]: Let Y be a nonempty fixed set An intuitionistic fuzzy set in Y is an object having the form = { y Y } where the functions :Y I and :Y I denotes the degree of membership (namely (y)) and the degree of non membership (namely υ (y)) of each element y Y to the set respectively, and 0 y Y 1 for each

On irresolute intuitionistic fuzzy multifunctions 1015 Definition 22 [1,2,3]: Let Y be a nonempty set and the intuitionistic fuzzy sets and be in the form = { : y Y}, = { : y Y} and let { : α } be an arbitrary family of intuitionistic fuzzy sets in Y Then: (a) if y Y and ]; (b) = if and ; (c) Ã c = { : y Y}; (d) 0 ={ y, 0,1 :y Y} and 1 ={ y,1,0 :y Y} (e) = { : y Y}; (f) = { : y Y}; Definition 23 [9] :Two Intuitionistic Fuzzy Sets and of Y are said to be quasi coincident ( q for short) if such that (y) or Lemma 21[9]: For any two intuitionistic fuzzy sets ( ) c and of Y, Definition 24[8]: An intuitionistic fuzzy topology on a non empty set Y is a family of intuitionistic fuzzy sets in Y which satisfy the following axioms: (O 1 ) 0, 1, (O 2 ) 1 (O 3 ) 2, for any 1, 2 for any arbitrary family { :, } In this case the pair (Y, ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in, is known as an intuitionistic fuzzy open set in Y The complement c of an intuitionistic fuzzy open set is called an intuitionistic fuzzy closed set in Y Definition 25 [9]: Let X be a nonempty set and are two real numbers such that a fixed element in X If then, (a) is called an intuitionistic fuzzy point(ifp in short) in X, where denotes the degree of membership of, and denotes the degree of non membership of

1016 S S Thakur and K Bohre (b) is called a vanishing intuitionistic fuzzy point (VIFP in short ) in X, where denotes the degree of non membership of Definition 25 [8]: Let (Y, be an intuitionistic fuzzy topological space and be an intuitionistic fuzzy set in Y Then the interior and closure of are defined by: cl( ) = { : is an intuitionistic fuzzy closed set in Y and }, int( ) = { : is an intuitionistic fuzzy open set in Y and } Lemma 22 [7]: For any intuitionistic fuzzy set in (Y, ) we have: (a) is an intuitionistic fuzzy closed set in Y (b) is an intuitionistic fuzzy open set in Y (c), (d) Definition 26[13]: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (Y, ) is called : (a) Intuitionistic fuzzy Semi-open if A (b) Intuitionistic fuzzy Semi-closed if its complement A c is Intuitionistic fuzzy semi open The family of all Intuitionistic fuzzy semi open (resp Intuitionistic fuzzy semi closed) sets of an intuitionistic fuzzy topological space (Y, is denoted by IFSO(Y) (resp IFSC(Y)) The intersection of all intuitionistic fuzzy semi-closed sets of Y containing A is called the semi-closure [13] of A, it is denoted by scla The union of all Intuitionistic fuzzy semi open subsets of A is called the semi interior of A, It is denoted by sint(a) An intuitionistic fuzzy set N of an intuitionistic fuzzy topological space(y, is called a semi neighborhood of an intuitionistic fuzzy point of Y if there exists an intuitionistic fuzzy semi open set O of Y such that N A is an intuitionistic fuzzy semi open set in Y if and only if it is a neighbourhood of each of its intuitionistic fuzzy points Definition 27 [24]: Let X and Y are two non empty sets A function F: X Y is called intuitionistic fuzzy multifunction if F(x) is an intuitionistic fuzzy set in Y, x X Definition 28[26]: Let F: X Y is an intuitionistic fuzzy multifunction and A be a subset of X Then F(A) =

On irresolute intuitionistic fuzzy multifunctions 1017 Lemma 23[26 ]: Let F : X Y be an intuitionistic fuzzy multifunction Then (a) A B F(A) F(B) for any subsets A and B of X (b) F(A B) F(A) F(B) for any subsets A and B of X (c) F( ) = {F ( ) : } for any family of subsets{ : } in X Definition 29[24]: Let F :X Y be an intuitionistic fuzzy multifunction Then the upper inverse F ( ) and lower inverse F ( ) of an intuitionistic fuzzy set in Y are defined as follows: F ( ) ={x : F(x) } F ( ) = {x : F(x)q } Lemma 24[26 ]: Let F : X Y be an intuitionistic fuzzy multifunction and, be intuitionistic fuzzy sets in Y Then: (a) F ( 1 ) = F ( 1 ) = X (b) F ( ) F ( ) (c) [F ( )] c = F ( c ) (d) [F ( )] c = F ( c ) (e) If then (f) If then Definition 210[24]: An Intuitionistic fuzzy multifunction F:X Y is said to be (a) Intuitionistic fuzzy upper semi continuous at a point x 0 X, if for any Intuitionistic fuzzy open set W Y such that F( x 0 ) W there exists an open set U X containing x0 such that F( U) W (b) Intuitionistic fuzzy lower semi continuous at a point x 0 X if for any Intuitionistic fuzzy open set W Y such that F( x 0 ) qw there exists an open set U X containing x0 such that F( x) qw, x U (c) Intuitionistic fuzzy upper semi-continuous (Intuitionistic fuzzy lower semicontinuous) if it is intuitionistic fuzzy upper semi-continuous (Intuitionistic fuzzy lower semi-continuous) at each point of X 3 Lower Irresolute Intuitionistic Fuzzy Multifunctions Definition 31: An Intuitionistic fuzzy multifunction F: ( is said to be:

1018 S S Thakur and K Bohre (a) Intuitionistic fuzzy lower irresolute at a point x 0 X if for any W IFSO( Y) such that F( x 0 ) qw there exists containing such that F( x) qw, x U (b) Intuitionistic fuzzy lower irresolute if it has this property at each point of X Definition 32: Let A be an intuitionistic fuzzy set of an intuitionistic fuzzy topological space ( ) Then V is said to be a semi-neighbourhood of A in Y if there exists an intuitionistic fuzzy semi-open set such that Theorem31: Let F: ( be an intuitionistic fuzzy multifunction and let Then the following statements are equivalent: (a) F is intuitionistic fuzzy lower irresolute at (b) For each with, implies ] (c) For any open set containing and for any with, there exists a non empty open set such that,, Proof : (a) (b) Let and such that, there is a,such that and Thus Now implies Hence (b) (c) Let such that, then Let be any open set such that then Put,then V is an open set, and (c) Let { } be the system of the open-sets in X containing x For any open set such that and any such that, there exists a non empty open set such that Let W =, then W is open, and Put S = W, then Thus, and hence F is intutionistic fuzzy lower irresolute at x Definition 33 [27]: Let X and Y are two non empty sets A function F: X Y is called fuzzy multifunction if F(x) is a fuzzy set in Y, x X Corollary 31[27]: For a fuzzy multifunction F:X Y and a point the following statements are equivalent : (a) F is fuzzy lower irresolute at (b) For each with, implies ) (c) For any open set containing and for any with, there exists a non empty open set such that, Corollary 32[25]: For a multifunction F:X Y and point the following statements are equivalent : (a) F is lower irresolute at (b) For each with, implies ))

On irresolute intuitionistic fuzzy multifunctions 1019 (c) For any open set containing and for any with, there exists a non empty open set such that, Theorem 32: Let F: ( be an intuitionistic fuzzy multifunction, Then the following statements are equivalent: (a) F is intuitionistic fuzzy lower irresolute (b) (c) For any and for each point,with, there is such that and (d) (e), for each intuitionistic fuzzy set, (f), for each subset (g),for each subset (h), for each Intuitionistic fuzzy set, Proof: (a) (b) Let and, so, since F is Intuitionistic Fuzzy lower irresolute, by Theorem 31 it follows that ) As x is chosen arbitrarily in we have ) and thus (b) (a) Let x be arbitrarily chosen in X and such that, so By hypothesis X), we have and thus F is intuitionistic fuzzy lower irresolute at x according to Theorem 31 As x was arbitrarily chosen, F is intuitionistic fuzzy lower irresolute (b) (c) Let and such that, thus Let, then and (c) (b) For any such that, there is a set such that and Then and hence (b) (d) Follows from lemma [24], (d) (e) Let be arbitrary intuitionistic fuzzy set of Y since by hypothesis ) Since we always have, by theorem 1[6], we obtain (e) (f) Suppose that (e) holds, and let A be an arbitrary subset of X Let us put then Therefore, by hypothesis, we have Therefore (f) (d) Suppose that (f) holds, and let Put ), then Therefore, by hypothesis, we have

1020 S S Thakur and K Bohre and Since we always have, we obtain Then by theorem 1[6], (d) (g) Since, we have Now scl(f(a)) and by hypothesis Thus Consequently, (g) (d) Let Replacing A by ) we get by(g) Consequently, But and so Thus by Theorem 3[6], (g) (h) Let be any intuitionistic fuzzy set of Y replacing A by we get by (g) Thus (h) (g) Replacing by F(A), where A is a subset of X, we get by (h) Thus the theorem This completes the proof of Corollary 33[27]: For a fuzzy multifunction F:X Y the following statements are equivalent : (a) F is fuzzy lower irresolute (b) (c) For any and for each point,with, there is such that and (d) (e), for each fuzzy set, (f), for each subset (g),for each subset (h), for each fuzzy set, Corollary 34[25]: For a multifunction F:X Y the following statements are equivalent : (a) F is lower irresolute (b) (c) For any and for each point,with, there is such that and (d) (e), for each set, (f), for each subset (g),for each subset (h), for each set,

On irresolute intuitionistic fuzzy multifunctions 1021 4 Upper Irresolute Intuitionistic Fuzzy Multifunctions Definition 41: An Intuitionistic fuzzy multifunction F: Y( is said to be (a) intuitionistic fuzzy Upper irresolute at a point x 0 X if for any such that F( x 0 ) W there exists containing x0 such that F( U) W (b) intuitionistic fuzzy upper irresolute if it has this property at each point of X Theorem 41: For an intuitionistic fuzzy multifunction F: ( and a point the following conditions are equivalent: (a) F is intuitionistic fuzzy upper irresolute at x (b) For each, there results the relation (c) For any open set containing x and for any, there exists a non-empty open set such that Proof (a) (b) Let and such that, there is a such that and Thus Since, Consequently, (b) (c) Let such that, then Let be any open set such that then Put, then V is an open set, and (c) (a) Let { } be the system of the open sets in X containing x For any open set such that and any such that, there exists a non empty open set such that Let W =, then W is open, and Put S = W, then Thus, and Hence F is intuitionistic fuzzy upper irresolute at x Corollary 41[27]: For an fuzzy multifunction F: Y( and a point the following conditions are equivalent: (a) F is fuzzy upper irresolute at x (b) For each, there results the relation (c) For any open set containing x and for any,there exists a non-empty open set such that

1022 S S Thakur and K Bohre Corollary 42[25]: For a multifunction F: Y( and a point the following conditions are equivalent: (a) F is upper irresolute at x (b) For each, there results the relation (c) For any open set containing x and for any,there exists a non-empty open set such that Theorem 42: For an intuitionistic fuzzy multifunction F: ( the following conditions are equivalent: (a) F is intuitionistic fuzzy upper irresolute (b) (c) For and with, there is a such that and (d) for each (e) For and for each semi-neighborhood of is a semineighborhood of x (f) For and for each semi-neighborhood of, there is a semineighborhood U of x such that (g) for each intuitionistic fuzzy set Proof: The proofs of (a) (b) (c) (d) are similar to the proof of Theorem 32 The equivalence of (d) and (g) are trivial (b) Let and be a semi-neighborhood of Then there is such that Hence Now by hypothesis and thus is a semi-neighborhood of x (e) Let and be a semi-neighborhood of Thus is a semi-neighborhood of x and (f) Let and such that, being a semi-open set, is a semi-neighborhood of F(x) and according to the hypothesis there is a semi-neighborhood U of x such that Therefore there is such that and hence Corollary 43[27]: For a fuzzy multifunction F: ( the following conditions are equivalent: (a) F is fuzzy upper irresolute (b) (c) For and with, there is a such that and (d) For each (e) For and for each semi-neighborhood V of is a semineighborhood of x

On irresolute intuitionistic fuzzy multifunctions 1023 (f) For and for each semi-neighborhood V of, there is a semineighborhood U of x such that (g) For each intuitionistic fuzzy set Corollary 44[25]: For a multifunction F from a topological space to another topological space ( the following conditions are equivalent: (a) F is upper irresolute (b) (c) For and with, there is a such that and (d) For each (e) For and for each semi-neighborhood V of is a semineighborhood of x (f) For and for each semi-neighborhood V of, there is a semineighborhood U of x such that (g) For each subset Corollary 45[25]: For a single-valued mapping f : X Y the following are equivalent: (a) f is irresolute (b) (c) For any and with, there is such that and (d) and for each semi-neighborhood V of is a semineighborhood of x (e) and for each semi-neighborhood V of, there is a semineighborhood U of x such that (f) For each (g) For each subset (h) For each subset (i) For each subset (j) For each subset 5 Irresolute Intuitionistic Fuzzy Multifunctions Definition 51: Let ( be an intuitionistic fuzzy topological space, an intuitionistic fuzzy set is called a semi q-neighbourhood of an intuitionistic fuzzy set of Y if such that Theorem 51: Let F: ( be an intuitionistic fuzzy multifunction, Then the following statements are equivalent: (a) F is intuitionistic fuzzy irresolute at a point ;

1024 S S Thakur and K Bohre (b) for any such that and,there result the relation (c) for every such that and, and for any open set U of X containing x, there exists a non-empty open set G U U, such that and Proof (a) (b): Let with and,there exists U SO(X) containing x such that and Thus, Since and, Therefore By theorem 1[15] we have (b) (c): Let with and Then Let be any open subset of X containing x Then U Since,, then Put, then, G U U,G U and G U And thus and (c) (a): Let be the family of open sets of X containing x For any open set containing x and for every with and, there exists a non-empty open set G U U such that and Let W= { } Then W is open in X, x Cl(W), F(W) and F(w)q, for every w W Put S=W {x}, then W S Cl(W) thus, and Hence F is intuitionistic fuzzy irresolute at x Theorem 52: Let F: ( be an intuitionistic fuzzy multifunction, Then the following statements are equivalent: (a) F is intuitionistic fuzzy irresolute (b) (c) (d) for any pair of intuitionistic fuzzy sets (e) for any pair of intuitionistic fuzzy sets

On irresolute intuitionistic fuzzy multifunctions 1025 (f), for any pair of intuitionistic fuzzy sets (g) For each point x of X for each semi-neighbourhood of F(x) and for each Semi q-neighbourhood of F(x) such that, is a semi-neighbourhood of x (h) For each point x of X and for each semi-neighbourhood of F(x) and for each Semi q-neighbourhood of F(x), such that, there is a semineighbourhood U of x such that and Proof: (a) (b) Let any and, thus and, Since F being intuitionistic fuzzy irresolute according to the theorem 51(b)There follows that And as x is chosen arbitrarily in, we have 1[15] and thus by Theorem (b) (c) It follows from Lemma 24 (c) and (d) (c) (d)suppose that (c) holds and let, Then Therefore by (c) By theorem 1[22] Now and By Lemma 24 (e) and (f) similarly and Consequently (e) Suppose that (d) hold Since A of X, it follows that for each subset (e) (f)

1026 S S Thakur and K Bohre And thus (f) (g) Let a semi-neighbourhood of F(x) and a semi q- neighbourhood of F(x), Then such that Therefore, By hypothesis It follows that is semi-neighbourhood of x (g) (h) Let a semi-neighbourhood of F(x) and a semi q- neighbourhood of F(x) Then by hypothesis, put Then is semi-neighbourhood of x, and ) (h) (a) Obvious Definition 52: An intuitionistic fuzzy multifunction F: ( is called : (a) intuitionistic fuzzy strongly lower semi-continuous for each intuitionistic fuzzy set (b) intuitionistic fuzzy strongly upper semi-continuous if X if for each intuitionistic fuzzy set is a open set in X if is a open set in Theorem 53: Let F: ( be an intuitionistic fuzzy upper irresolute and intuitionistic fuzzy strongly lower semi-continuous intuitionistic fuzzy multifunction then F is intuitionistic fuzzy irresolute Proof: Let and let Now F being intuitionistic fuzzy upper irresolute, there is a semi-open set U of X containing x such that Since U is semi-open in X we have, ))) As x is chosen arbitrarily in ) there follows that )) and thus by theorem1 of [15] Again F being intuitionistic fuzzy strongly lower semi-continuous, ) is an open set in X Hence, ) and by Theorem 52, F is intuitionistic fuzzy irresolute References [1] K Atanassov, Intuitionistic Fuzzy Sets, In VII ITKR s Session, (V Sgurev, Ed) Sofia, Bulgaria, (1983)

On irresolute intuitionistic fuzzy multifunctions 1027 [2] K Atanassov and S Stoeva, Intuitionistic Fuzzy Sets, In Polish Symposium on Interval and Fuzzy Mathematics, Poznan, (1983), 23-26 [3] K Atanassov, Intuitionistic Fuzzy Sets Fuzzy Sets and Systems, 20(1986), 87-96 [4] Bayhan Sadik, On Separation Axioms in Intuitionistic Topological Spaces Intern Jour Math Math Sci 27(10)(2001), 621-630 [5] C Berge, Espaces topologiquesfonctions multivoques Dunod Paris1959 [6] N Biswas, On characterizations of Semi-continuous functions Atti della Accademia Nazionale dei Lincei, Rendiconti, Classe di Scienze Fisiche, Matematiche e Naturali 47(8)(1970): 399-402 [7] C L Chang, Fuzzy Topological Spaces, J Math Anal Appl 24(1968), 182-190 [8] D Coker, An Introduction to Intuitionistic Fuzzy Topological Spaces, Fuzzy Sets and Systems 88(1997), 81-89 [9] D Coker and M Demirci, On Intuitionistic Fuzzy Points Notes on Intuitionistic Fuzzy Sets 2(1)(1995), 78-83 [10] D Coker and A Es Hayder, On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces, The Journal of Fuzzy Mathematics (3-4}(1995), 899-909 [11] S G Crossley, and S K Hildenbrand, Semi topological properties Fundamenta Mathematicae: 74(1972); 233-54 [12] J Evert and T Lipski, Quasi-continuous multivalued mappings Mathematica Slovaca 33(1983):69-74 [13] H Gurcay, D Coker and A Es Hayder, On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces The Journal of Fuzzy Mathematics 5(2)(1997), 365-378 [14] T R Hamlett, Semi-continuous and irresolute functions Texas Journal of Science 27(1976): 45-49 [15] N Levine, semi-open sets and semi-continuity in topological spaces, Amer Math Monthly, 70(1963), 36-41 [16] F G Lupianez Separation axioms in intuitionistic fuzzy topological spaces, International Journal of Pure and Applied Mathematics 17(1)(2004) 29-34 [17] F G Lupianez Nets and filters in intuitionistic fuzzy topological spaces, Information Sciences 176(2006)2396-2404 [18] F G Lupianez Housdorffness in intuitionistic fuzzy topological Spaces, The Journal of Fuzzy Mathematics 12(3)(2004)521-525 [19] S N Maheshwari, G I Chae and S S Thakur, Some new mappings in Bitopological spaces Ulsan Institute of Technology Report 12(2)(1981):301-04 [20] S N Maheshwari and R Prasad On pairwise irresolute functions Mathematica,18(2)(1976), 169-72 [21] A Neubrunnov, On certain generalizations of the notions of continuity Mathematiki Casopis 23(4)(1973),374-80

1028 S S Thakur and K Bohre [22] T Noiri, A note on semi-continuous mapping Atti della Accademia Nazionale dei Lincei, Rendiconti, Classe di Scienze Fisiche, Matematiche e Naturali 55(1973): 400-03 [23] T Noiri Semi-continuity and weak continuity Czechoslovak Mathematical journal 31(1981):314-21 [24] O Ozbakir and D Coker, Fuzzy Multifunction s in Intuitionistic Fuzzy Topological Spaces Notes on Intuitionistic Fuzzy Sets 5(3)(1999) [25] V Popa, On Characterizations of irresolute multifunctions JUniv Kuwait (sci)(15)(1988) 21-25 [26] S S Thakur and Kush Bohre, On intuitionistic fuzzy multifunctions (submitted) [27] V Seenivasan and G Balasubramanian, On upper and lower semi irresolute fuzzy multifunctions, East Asian mathj23 (2007)(1):9-22 [28] T Thompson, Semi-continuous and irresolute images of S-closed spaces Proceedings of the American Mathematical Society 66(2)(1977):359-62 [29] B Tripathy, Intuitionistic Fuzzy Metric Spaces, Notes on Intuitionistic Fuzzy Sets 5(2)(1999) [30] N Turnali and D Coker, Fuzzy Connectedness in Intuitionistic Fuzzy Topological Spaces Fuzzy Sets and Systems 116 (3) (2000), 369-375 [31] L A Zadeh, Fuzzy Sets, Information and Control, 18(1965), 338-353 Received: December, 2010