Strongly intuitionistic fuzzy rw-continuous functions

In this paper, we introduce the strongly intuitionistic fuzzy rw-continuous functions and study its properties. Furthermore we introduced perfectly intuitionistic fuzzy rw-continuous functions, intuitionistic fuzzy rw-irresolute, and intuitionistic fuzzy Trw-space in intuitionistic fuzzy topological spaces.

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Introduction

After the introduction of fuzzy sets by Zadeh [22] in 1965 and fuzzy topology by Chang [3] in1967, several researches were conducted on the generalizations of the notions of fuzzy sets andfuzzy topology. The concept of intuitionistic fuzzy sets (IFS) was introduced by Atanassov [1] as a generalization of fuzzy sets. In the last 25 years various concepts of fuzzy mathematics have been extended for IFS. In 1997 Coker [4] introduced the concept of IF topological spaces. Recently many fuzzy topological concepts such as fuzzy compactness [6] , fuzzy connectedness [20] , fuzzy separation axioms [2] , fuzzy continuity [7] , fuzzy g-closed sets [10] , fuzzy g-continuity [11] , fuzzy rg-closed sets [12] have been generalized for IF topological spaces. Recently authors of this paper introduced the concept of IF w-closed sets [16] in IF topology. S.S. Thakur and Jyoti Pandey Bajpai [17] introduced intuitionistic fuzzy rw-continuity. In the present paper we introduced strongly intuitionistic fuzzy rw-continuous functions and also introduced perfectly intuitionistic fuzzy rw-continuous functions. We obtain some of their characterization and properties.

Preliminaries

Let X is a nonempty fixed set. An intuitionistic fuzzy set A [1] in X is an object having the form A = {<x, μA(x), γA(x)> : x∈ X}, where the functions μA: X→[0,1] and γA: X→[0,1] denotes the degree of membership μA(x) and the degree of non-membership γA(x) of each element x∈X to the set A respectively and 0 ≤ μA(x)+ γA(x) ≤ 1 for each x∈X. The intuitionistic fuzzy sets 0̃ = {< x, 0, 1> : x ∈ X} and ĩ = {<x, 1, 0> : x ∈ X} are respectively called empty and whole intuitionistic fuzzy set on X. An intuitionistic fuzzy set A = {<x, μA(x), γA(x)> : x ∈ X} is called a subset of an intuitionistic fuzzy set B = {<x, μA(x), γB(x)> : x ∈ X} (for short A ⊆ B) if μA(x) ≤ μA(x) and γA(x)≥ μA(x) for each x ∈ X. The complement of an intuitionistic fuzzy set A = {<x, μA(x), γA(x)> : x ∈ X} is the intuitionistic fuzzy set A c ={<x, γA(x), μA(x) >: x ∈ X}. The intersection (resp. union) of any arbitrary family of intuitionistic fuzzy sets Ai = {< x, μA(x), γA(x) > : x ∈ X, ( i∈∧)} of X be the intuitionistic fuzzy set ∩Ai ={< x, μA(x), ∨ γAi(x) > : x ∈ X} (resp. ∪Ai ={<x, ∨ μA(x), ∧ γAi(x) >: x ∈ X}). Two intuitionistic fuzzy sets A = {<x, μA(x), γA(x)> : x ∈ X} and B = {<x, μB(x), γB(x)> : x ∈ X} are said to be q-coincident (AqB for short) if and only if ∃ an element x∈ X such that μA(x) > γB(x)or γA(x)< μB(x). A family τ of intuitionistic fuzzy sets on a non-empty set X is called an intuitionistic fuzzy topology [3] on X if the intuitionistic fuzzy sets õ, ĩ∈ τ, and ℑ is closed under arbitrary union and finite intersection. The ordered pair (X, τ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in τ is called an intuitionistic fuzzy open set.

The compliment of an intuitionistic fuzzy open set in X is known as intuitionistic fuzzy closed set. The intersection of all intuitionistic fuzzy closed sets which contains A is called the closure of A. It denoted by cl (A). The union of all intuitionistic fuzzy open subsets of A is called the interior of A. It is denoted by int (A) [4] . Definition 2.1 [5] : Let X is a nonempty set and c∈X a fixed element in X. If α∈(0,1] and β∈[0,1) are two real numbers such that α+β≤1 then : a) c(α,β) = < x,cα, c1-β > is called an intuitionistic fuzzy point in X, where α denotes the degree of membership of c(α,β), and β denotes the degree of non-membership of c(α, β). b) c(β) = < x,0, 1-c1-β > is called a vanishing intuitionistic fuzzy point in X, where β denotes the degree of nonmembership of c(β).

Definition 2.2 [6] : A family {Gi : i∈∧} of intuitionistic fuzzy sets in X is called an intuitionistic fuzzy open cover of X if ∪{Gi : i∈∧} = ĩ and a finite subfamily of an intuitionistic fuzzy open cover {Gi: i∈∧}of X which also an intuitionistic fuzzy open cover of X is called a finite sub cover of {Gi: i∈∧}.

Definition 2.3 [6] : An intuitionistic fuzzy topological space (X,τ) is called intuitionistic fuzzy compact if every intuitionistic fuzzy open cover of X has a finite sub cover.

Definition 2.4: An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, τ) is called: a) Intuitionistic fuzzy g-closed [10] if cl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open in X. b) Intuitionistic fuzzy sg-closed [15] if scl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy semi open in X. c) Intuitionistic fuzzy rg-closed [12] if cl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy regular open in X. d) Intuitionistic fuzzy αg-closed [8] if αcl(A) ⊆ O whenever A ⊆O and O is intuitionistic fuzzy open in X. e) Intuitionistic fuzzy gsp-closed [9] if spcl(A) ⊆ O whenever A ⊆O and O is intuitionistic fuzzy open in X. f) Intuitionistic fuzzy w-closed [16] if cl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy semi open in X. g) Intuitionistic fuzzy rw-closed [17] if cl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy regular semi open in X. h) Intuitionistic fuzzy gpr-closed [18] if pcl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open in X. i) Intuitionistic fuzzy rgα-closed [19] if αcl(A) ⊆ O whenever A ⊆O and O is intuitionistic fuzzy regular αopen in X. The complements of the above mentioned closed sets are their respective open sets. Definition 2.5 [10] : An intuitionistic fuzzy topological space (X, τ) is said to be intuitionistic fuzzy-T1/2 if every intuitionistic fuzzy g-closed set in X is intuitionistic fuzzy closed in X. Definition 2.6 [19] : An intuitionistic fuzzy topological space (X, τ) is said to be intuitionistic fuzzy rgα-T1/2 if every intuitionistic fuzzy rgα-closed set in X is intuitionistic fuzzy closed in X. Definition 2.7 [7] : Let (X, τ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f : X→Y be a function. Definition 2.8: Let (X, τ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f: X→Y be a function. Then f is said to be: a) Intuitionistic fuzzy g-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy g -closed in X [11] . b) Intuitionistic fuzzy sg-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy sg -closed in X [16] . c) Intuitionistic fuzzy αg-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy αg -closed in X [8] . d) Intuitionistic fuzzy rg-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy rg -closed in X [13] . e) Intuitionistic fuzzy rg-irresolute if pre image of every intuitionistic fuzzy rg-closed set in Y is intuitionistic fuzzy rg -closed in X [14] . f) Intuitionistic fuzzy w-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy w -closed in X [16] . g) Intuitionistic fuzzy gpr-continuous if pre image of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy gpr -closed in X [18] .

Remark 2.1 a) Every intuitionistic fuzzy continuous mapping is intuitionistic fuzzy w-continuous [16] . b) Every intuitionistic fuzzy w-continuous mapping is intuitionistic fuzzy g-continuous [16] . c) Every intuitionistic fuzzy g-continuous mapping is intuitionistic fuzzy rg-continuous [13] . d) Every intuitionistic fuzzy g-continuous mapping is intuitionistic fuzzy gpr-continuous [18] . e) Every intuitionistic fuzzy rg-continuous mapping is intuitionistic fuzzy gpr-continuous [18] . The converse of above statements may be false.

Strongly Intuitionistic Fuzzy rw-Continuous functions Definion: 3.1

A function f: (X,τ )→(Y, σ) is called strongly instuitionistic fuzzy rw-continuous in IFTS if f -1 (V) is instuitionistic fuzzy closed in X for every instuitionistic fuzzy rw-closed set V in Y.

Example 3.1

Let X={a,b},Y={x,y} and instuitionistic fuzzy sets U and V are defined as follows, U=(<a,0,0.2>,<b,0.4,0.2>) V=(<a,0.4,0.1>,<b,0.6,0.1>) and W=(<x,0.2,0>,<y,0.2,0.4>) Let τ={õ,ĩ,U,V} and σ ={õ,ĩ,W} be instuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:

1. Atanassov K, Stoeva S. Intuitionistic Fuzzy Sets, In Polish Symposium on Interval and Fuzzy Mathematics, Poznan, 1983, 23-26.
2. Bayhan S. On separation axioms in Intuitionistic Topological spaces. Intern. Jour. Math. Sci. 2001; 27(10):621-630.
3. Chang CL. Fuzzy Topological Spaces, J Math. Anal. Appl. 1968; 24:182-190.
4. Çoker D. An Introduction to Intuitionistic Fuzzy Topological Spaces, Fuzzy Sets and Systems. 1997; 88:81-89.
5. Cocker D, Demirci M. On Fuzzy Points notes on IFS. 1995; 2:78-83.
6. Çoker DA. Es. Haydar. On Fuzzy Compactness in Intuitionistic Fuzzy Topological Spaces. The Journal of Fuzzy Mathematics. 1995; 3(4):899-909.
7. Gurcay HD, Çoker A, Es. Haydar. On Fuzzy Continuity in Intuitionistic Fuzzy Topological Spaces. The Journal of Fuzzy Mathematics. 1997; 5(2):365-378.
8. Sakthivel K. Intuitionistic fuzzy Alpha generalized irresolute mappings, Applied mathematical sciences. 2010; 4(37):1831-1842.
9. Santi, Jayanthi D. Intuitionistic Fuzzy generlalized semi Pre Continuous Mappings, Int, J Contemp. Math. Sciences. 2010; 5(30):1455-1469.
10. Thakur SS, Chaturvedi R. Generalized closed set in intuitionistic fuzzy topology. The Journal of Fuzzy Mathematics. 2008; 16(3):559-572.
11. Thakur SS, Chaturvedi R. Generalized Continuity in intuitionistic fuzzy topological spaces. Notes on Intutionistic Fuzzy Sets. 2006; 12(1):38-44.
12. Thakur SS, Chaturvedi R. Regular generalized closed sets in intuitionistic fuzzy topology, Studii Si Cercetari Stiintifice Seria Mathematica. 2010; 16:257-272.
13. Thakur SS, Chaturvedi R. Intuitionistic fuzzy rg continuous mappings.Journal of Indian academy of mathematics. 2007; 29(2):467-473.
14. Thakur SS, Chaturvedi R. Intuitionistic fuzzy rgirresolute mappings, Varahamihira journal of Mathematical Sciences. 2006; 6(1):199-204.
15. Thakur SS, Bajpai Pandey Jyoti. Semi Generalized closed set in intuitionistic fuzzy topology, 2006.
16. Thakur SS, Bajpai Pandey Jyoti. Intuitionistic Fuzzy wclosed sets and intuitionistic fuzzy w-continuity, International Journal of Contemporary Advanced Mathematics. 2010; 1(1):1-15.
17. Thakur SS, Bajpai Pandey Jyoti. Intuitionistic fuzzy rgαclosed sets, International Journal of Fuzzy System and Rough System. 2011; 4(1):67-73.
18. Turnali N, Çoker D. Fuzzy Connectedness in Intuitionistic Fuzzy topological Spaces. Fuzzy Sets and Systems. 2000; 116(3):369-375.
19. Young Bae Jun, Seok Zun Song. Intuitionistic fuzzy semi pre open sets and intuitionistic fuzzy semi pre continuous mappings. jour. of App. Math. and Computing, 2005, 467-474.
20. Zadeh LA. Fuzzy Sets, Information and Control. 1965 18:338-353.