Generalized Semipre Regular Continuous and Irresolute Mappings in Intuitionistic Fuzzy Topological Space

Generalized Semipre Regular Continuous and Irresolute Mappings in Intuitionistic Fuzzy Topological Space

K. Ramesh

Department of Mathematics, NGM College, Pollachi-642001, Tamil Nadu, India.

M. Thirumalaiswamy

Department of Mathematics, NGM College, Pollachi-642001, Tamil Nadu, India.

S. Kavunthi

Department of Mathematics Shri Nehru Mahavidyalaya College

of Arts and Science, Coimbatore-641 050, Tamil Nadu, India.

Abstract

In this paper, we introduce and study the notions of intuitionistic fuzzy generalized semipre regular continuous mappings and intuitionistic fuzzy generalized semipre regular irresolute mappings and study some of its properties in intuitionistic fuzzy topological spaces.

Keywords: Intuitionistic fuzzy topology, Intuitionistic fuzzy point, Intuitionistic fuzzy generalized semipre regular closed sets, Intuitionistic fuzzy generalized semipre regular continuous mappings and Intuitionistic fuzzy generalized semipre regular irresolute mappings.

2010 Mathematics Subject Classification: 54A40, 03F55.

1. Introduction

   The concept of fuzzy set [FS] was introduced by Zadeh [20] and later fuzzy topology was introduced by Chang [3] in 1967. By adding the degree of non membership to FS, Atanassov [1] proposed intuitionistic fuzzy set [IFS] using the notion of fuzzy sets. On the other hand Coker [4] introduced intuitionistic fuzzy topological spaces using the notion of intuitionistic fuzzy sets. In this paper we introduce intuitionistic fuzzy generalized semipre regular continuous mappings and intuitionistic fuzzy generalized semipre regular irresolute mappings and studied some of their basic properties.

2. Preliminaries

   Throughout this paper, (X, ) or X denotes the intuitionistic fuzzy topological spaces (briefly IFTS). For a subset A of X, the closure, the interior and the complement of A are denoted by cl(A), int(A) and Ac respectively. We recall some basic definitions that are used in the sequel.

   Definition 2.1: [1]

   Let X be a non-empty fixed set. An intuitionistic fuzzy set (IFS in short) A in X is an object having the form A = {<x,

   µA(x), A(x)>/ x X} where the functions µA : X [0,1] and A : X [0,1] denote the degree of membership (namely µA(x)) and the degree of nonmembership (namely A(x)) of each element x X to the set A, respectively, and 0 µA(x) + A(x) 1 for each x X. Denote by IFS(X),

   the set of all intuitionistic fuzzy sets in X.

   Definition 2.2: [1]

   Let A and B be IFSs of the form A = {<x, µA(x), A(x)>/ x

   X} and B = {<x, µB(x), B(x)>/ x X}. Then

   1. A B if and only if µA(x) µB(x) and A(x) B(x) for all x X,

   2. A = B if and only if A B and B A,

   3. Ac = { <x, A(x), µA(x)>/ x X },

      (iv) A B = { <x, µA(x) µB(x), A(x) B(x)>/ x

      X },

      (v) A B = { <x, µA(x) µB(x), A(x) B(x)>/ x

      X }.

      For the sake of simplicity, we shall use the notation A = <x,

      µA, A> instead of A = {<x, µA(x), A(x)>/ x X }. Also for the sake of simplicity, we shall use the notation A = <x, (µA, µB), (A, B)> instead of A = <x, (A/A, B/B), (A/A,

      B/B)>. The intuitionistic fuzzy sets 0~ = { <x, 0, 1>/ x

      X } and 1~ = { <x, 1, 0>/ x X } are respectively the empty set and the whole set of X.

      Definition 2.3: [4]

      An intuitionistic fuzzy topology (IFT in short) on X is a family of IFSs in X satisfying the following axioms:

      (i) 0~, 1~ ,

      1. G1 G2 , for any G1, G2 ,

      2. Gi for any family {Gi / i J } .

         In this case the pair (X, ) is called an intuitionistic fuzzy topological space (IFTS in short) and any IFS in is known as an intuitionistic fuzzy open set(IFOS in short)in X. The complement Ac of an IFOS A in an IFTS (X, ) is called an intuitionistic fuzzy closed set (IFCS in short) in X.

         Definition 2.4: [4]

         Let (X, ) be an IFTS and A = <x, µA, A> be an IFS in X.

         Then

         1. int(A) = { G / G is an IFOS in X and G A},

         2. cl(A) = { K / K is an IFCS in X and A K},

         3. cl(Ac) = (int(A))c,

         4. int(Ac) = (cl(A))c.

      Definition 2.5: [11]

      An IFS A of an IFTS (X, ) is an

      1. intuitionistic fuzzy semiclosed set (IFSCS in short) if int(cl(A)) A,

      2. intuitionistic fuzzy semiopen set (IFSOS in short) if A cl(int(A)).

      Definition 2.6: [6]

      An IFS A of an IFTS (X, ) is an

      1. intuitionistic fuzzy preclosed set (IFPCS in short) if cl(int(A)) A,

      2. intuitionistic fuzzy preopen set (IFPOS in short) if A int(cl(A)).

      Note that every IFOS in (X, ) is an IFPOS in X.

      Definition 2.7: [6]

      An IFS A of an IFTS (X, ) is an

      1. intuitionistic fuzzy regular closed set (IFRCS in short) if A = cl(int(A)),

      2. intuitionistic fuzzy regular open set (IFROS in short) if A = int(cl(A)).

      Definition 2.8: [19]

      An IFS A of an IFTS (X, ) is an

      1. intuitionistic fuzzy semipre closed set (IFSPCS for

         short) if there exists an IFPCS B such that int(B) A B,

      2. intuitionistic fuzzy semipre open set (IFSPOS for short) if there exists an IFPOS B such that B A

      cl(B).

      Definition 2.9: [9]

      Let A be an IFS in an IFTS (X, ). Then

      1. spint (A) ={G / G is an IFSPOS in X and G

         A }.

      2. spcl (A) = {K / K is an IFSPCS in X and A K

      }.

      Note that for any IFS A in (X, ), we have spcl(Ac) = (spint(A))c and spint(Ac) = (spcl(A))c .

      Definition 2.10:

      Let A be an IFS in an IFTS (X, ). Then

      1. intuitionistic fuzzy regular generalized closed set

         (IFRGCS for short) if cl(A) U whenever A U and U is an Intuitionistic fuzzy regular open in X [12],

      2. intuitionistic fuzzy generalized pre regular closed

         set (IFGPRCS for short) if pcl(A) U whenever A U and U is an Intuitionistic fuzzy regular open in X [15],

      3. intuitionistic fuzzy generalized pre closed set

      (IFGPCS for short) if pcl(A) U whenever A U and U is an Intuitionistic fuzzy open in X [7].

      An IFS A of an IFTS (X, ) is called an intuitionistic fuzzy regular generalized open set, intuitionistic fuzzy generalized pre regular open set and intuitionistic fuzzy generalized pre open set (IFRGOS, IFGPROS and IFGPOS in short) if the complement Ac is an IFRGCS, IFGPRCS and IFGPCS respectively.

      Definition 2.11: [17]

      An IFS A in an IFTS (X, ) is said to be an intuitionistic fuzzy semipre generalized closed set (IFSPGCS for short) if spcl(A) U whenever A U and U is an IFSOS in (X, ).

      Definition 2.12: [16]

      The complement Ac of an IFSPGCS A in an IFTS (X, ) is called an intuitionistic fuzzy semipre generalized open set (IFSPGOS for short) in X.

      Definition 2.13: [9]

      An IFS A is an IFTS (X, ) is said to be an intuitionistic fuzzy generalized semipre closed set (IFGSPCS) if spcl(A)

      U whenever A U and U is an IFOS in (X, ). An IFS

      A of an IFTS (X, ) is called an intuitionistic fuzzy generalized semipre open set (IFGSPOS in short) if Ac is an IFGSPCS in (X, ).

      Definition 2.14: [8]

      An IFS A in an IFTS (X, ) is said to be an intuitionistic fuzzy generalized semipre regular closed set (IFGSPRCS for short) if spcl(A) U whenever A U and U is an IFROS in (X, ).

      Every IFCS, IFRCS IFPCS, IFSCS, IFSPCS, IFSPGCS, IFGSPCS, IFRGCS, IFGPRCS is an IFGSPRCS but the

      converses are not true in general.

      Definition 2.15: [8]

      The complement Ac of an IFGSPRCS A in an IFTS (X, ) is called an intuitionistic fuzzy generalized semipre regular open set (IFGSPROS for short) in X.

      The family of all IFGSPROSs of an IFTS (X, ) is denoted by IFGSPRO(X). Every IFOS, IFROS, IFPOS, IFSOS, IFSPOS, IFSPGOS, IFGSPOS, IFRGOS, IFGPROS is an

      IFGSPROS but the converses are not true in general.

      Definition 2.16: [11]

      Let , [0, 1] and + 1. An intuitionistic fuzzy point (IFP for short) (, ) of X is an IFS of X defined by

      (, ) ) = (, ) if y = p or (0, 1) if y p.

      Result 2.17: [2]

      For an IFS A in an IFTS(X, ), we have spcl(A) A

      (int(cl(int(A)))).

      Definition 2.18: [8]

      If every IFGSPRCS in (X, ) is an IFSPCS in (X, ), then the space can be called as an intuitionistic fuzzy semipre regular T1/2 (IFSPRT1/2 for short) space.

      Definition 2.19: [6]

      Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be intuitionistic fuzzy continuous (IF continuous in short) if f -1(B) IFO(X) for every B .

      Definition 2.20: [6]

      Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ).

      Then f is said to be an

      1. intuitionistic fuzzy semi continuous(IFS continuous in short) if f -1(B) IFSO(X) for every B ,

      2. intuitionistic fuzzy pre continuous(IFP continuous in short) if f -1(B) IFPO(X) for every B .

      Definition 2.21: [19]

      Let f be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an intuitionistic fuzzy semipre continuous (IFSP continuous for short) mapping if f -1(B)

      IFSPO(X) for every B .

      Every IFS continuous mapping and IFP continuous mappings are IFSP continuous mapping but the converses may not be true in general.

      Definition 2.22: [14]

      A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy regular generalized continuous (IFRG continuous for short) mappings if f -1(V) is an IFRGCS in (X, ) for every IFCS V of (Y, ).

      Definition 2.23: [13]

      A mapping f: (X, ) (Y, ) be an intuitionistic fuzzy regular generalized irresolute (IFRG irresolute) mapping if f -1(V) is an IFRGCS in (X, ) for every IFRGCS V of (Y, ).

      Definition 2.24: [15]

      A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized pre regular continuous (IFGPR continuous for short) mappings if f -1(V) is an IFGPRCS in (X, ) for every IFCS V of (Y, ).

      Definition 2.25: [18]

      A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy semipre generalized continuous (IFSPG continuous for short) mappings if f -1(V) is an IFSPGCS in (X, ) for every IFCS V of (Y, ).

      Definition 2.26: [10]

      A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized semipre continuous (IFGSP continuous for short) mapping if f -1(V) is an IFGSPCS in (X, ) for every IFCS V of (Y, ).

      Result 2.27: [8]

      For any IFS A in (X, ) where X is an IFSPRT1/2 space, A IFGSPRO(X) if and only if for every IFP p(, ) A, there exists an IFGSPROS B in X such that p(, ) B A.

3. Intuitionistic Fuzzy Generalized Semipre Regular Continuous Mappings

   Govindappa Navalagi, A. S. Chandrashekarappa and S. V. Gurushantanavar [5] have introduced GSPR-continuous mappings in topological spaces. In this section we have introduced intuitionistic fuzzy generalized semipre regular continuous mapping and investigated some of its properties in intuitionistic fuzzy topological spaces.

   Definition 3.1:

   A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized semipre regular continuous (IFGSPR continuous for short) mappings if f -1(V) is an IFGSPRCS in (X, ) for every IFCS V of (Y, ).

   For the sake of simplicity, we shall use the notation A= <x, (µ, µ), (, )> instead of A= <x, (a/a, b/b), (a/a, b/b)> in all the examples used in this paper.

   Example 3.2:

   Let X= {a, b}, Y= {u, v} and G1 = <x, (0.5, 0.4), (0.5, 0.6)>,

   G2 = <y, (0.6, 0.7), (0.4, 0.2)>. Then = {0~, G1, 1~} and =

   {0~, G2, 1~} are IFTs on X and Y respectively. Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping.

   Theorem 3.3:

   Every IF continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IF continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFCS in X. Since every IFCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.4:

   In Example 3.2, f: (X, ) (Y, ) is an IFGSPR continuous mapping but not an IF continuous mapping. Since G2 = <y, (0.6, 0.7), (0.4, 0.2)> is an IFOS in Y but f -1(G2) = <x, (0.6,

   0.7), (0.4, 0.2)> is not an IFOS in X.

   Theorem 3.5:

   Every IFS continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFS continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFSCS in X. Since every IFSCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.6:

   In Example 3.2, f: (X, ) (Y, ) is an IFGSPR continuous mapping but not an IFS continuous mapping. Since G2 =

   <y, (0.6, 0.7), (0.4, 0.2)> is an IFOS in Y but f -1(G2) = <x,

   (0.6, 0.7), (0.4, 0.2)> is not an IFSOS in X.

   Theorem 3.7:

   Every IFP continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFP continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFPCS in X. Since every IFPCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.8:

   Let X = {a, b}, Y = {u, v} and G1 = <x, (0.3, 0.2), (0.6,

   0.6)>, G2 = <y, (0.4, 0.5), (0.6, 0.5)>. Then = {0~, G1, 1~}

   and = {0~, G2, 1~} are IFTs on X and Y respectively.

   Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping but not an IFP continuous mapping.

   Theorem 3.9:

   Every IFSP continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFSP continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFSPCS in X. Since every IFSPCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.10:

   Let X = {a, b}, Y = {u, v} and G1 = <x, (0.7, 0.8), (0.3,

   0.2)>, G2 = <x, (0.2, 0.1), (0.8, 0.9)>, G3 = <x, (0.5, 0.6),

   (0.5, 0.4)>, G4 = <x, (0.6, 0.7), (0.4, 0.3)>, G5 = <y, (0.1, 0.4), (0.9, 0.6)>. Then = {0~, G1, G2, G3, G4, 1~} and =

   {0~, G5, 1~} are IFTs on X and Y respectively. Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping but not an IFSP continuous mapping.

   Theorem 3.11:

   Every IFRG continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFRG continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFRGCS in X. Since every IFRGCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.12:

   Let X = {a, b}, Y = {u, v} and G1 = <x, (0.5, 0.4), (0.5,

   0.6)>, G2 = <y, (0.6, 0.7), (0.4, 0.2)>. Then = {0~, G1, 1~}

   and = {0~, G2, 1~} are IFTs on X and Y respectively. Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping but not an IFRG continuous mapping.

   Theorem 3.13:

   Every IFGPR continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFGPR continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFGPRCS in X.Since every IFGPRCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.14:

   Let X = {a, b}, Y = {u, v} and G1 = <x, (0.7, 0.8), (0.3,

   0.2)>, G2 = <x, (0.2, 0.1), (0.8, 0.9)>, G3 = <x, (0.5, 0.6),

   (0.5, 0.4)>, G4 = <x, (0.6, 0.7), (0.4, 0.3)> and let G5 = <y,

   (0.3, 0.2), (0.7, 0.8)>. Then = {0~, G1, G2, G3, G4, 1~} and

   = {0~, G5, 1~} are IFTs on X and Y respectively. Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping but not an IFGPR continuous mapping.

   Theorem 3.15:

   Every IFSPG continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFSPG continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFSPGCS in X. Since every IFSPGCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.16:

   Let X = {a, b}, Y = {u, v} and G1 = <x, (0.3, 0.6), (0.7,

   0.4)>, G2 = <y, (0.7, 0.4), (0.3, 0.6)>. Then = {0~, G1, 1~}

   and = {0~, G2, 1~} are IFTs on X and Y respectively. Define a mapping f: (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPR continuous mapping but not an IFSPG continuous mapping.

   Theorem 3.17:

   Every IFGSP continuous mapping is an IFGSPR continuous mapping but not conversely.

   Proof: Let f: (X, ) (Y, ) be an IFGSP continuous mapping. Let V be an IFCS in Y. Then f -1(V) is an IFGSPCS in X. Since every IFGSPCS is an IFGSPRCS, f -1(V) is an IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   Example 3.18:

   In Example 3.16, f: (X, ) (Y, ) is an IFGSPR continuous mapping but not an IFGSP continuous mapping.

   Since G2 = <y, (0.7, 0.4), (0.3, 0.6)> is an IFOS in Y but f -1(G2) = <x, (0.7, 0.4), (0.3, 0.6)> is not an IFGSPOS in X.

   Theorem 3.19:

   Let f: (X, ) (Y, ) be a mapping where f -1(V) is an IFRCS in X for every IFCS in Y. Then f is an IFGSPR continuous mapping but not conversely.

   Example 3.20:

   In Example 3.2, f: (X, ) (Y, ) is an IFGSPR continuous mapping but not a mapping defined in Theorem 3.19.

   Theorem 3.21:

   If f: (X, ) (Y, ) is an IFGSPR continuous mapping, then for each IFP p(, ) of X and each A such that f(p(, ))

   A. Then there exists an IFGSPROS B of X such that p(, )

   B and f(B) A.

   Proof: Let p(, ) be an IFP of X and A such that f(p(, ))

   A. Put B = f -1(A). Then by hypothesis B is an IFGSPROS in X such that p(, ) B and f(B) = f(f -1(A)) A.

   Theorem 3.22:

   Let f: (X, ) (Y, ) be an IFGSPR continuous mapping. Then f is an IFSP continuous mapping if X is an IFSPRT1/2 space.

   Proof: Let V be an IFCS in Y. Then f -1(V) is an IFGSPRCS in X, by hypothesis. Since X is an IFSPRT1/2 space, f -1(V) is an IFSPCS in X. Hence f is an IFSP

   continuous mapping.

   Theorem 3.23:

   Let f: (X, ) (Y, ) be an IFGSPR continuous mapping and g: (Y, ) (Z, ) is an IF continuous mapping, then g f : (X, ) (Z, ) is an IFGSPR continuous mapping.

   Proof: Let V be an IFCS in Z. Then g -1(V) is an IFCS in

   Y, by hypothesis. Since f is an IFGSPR continuous mapping, f -1(g -1(V)) is an IFGSPRCS in X. Hence g f is an IFGSPR continuous mapping.

   Theorem 3.24:

   Let f: (X, ) (Y, ) be a mapping from an IFTS X into an IFTS Y. Then the following conditions are equivalent if X and Y are IFSPRT1/2 space:

   1. f is an IFGSPR continuous mapping,

   2. f -1(B) is an IFGSPROS in X for each IFOS B in Y,

   3. for every IFP p(, ) in X and for every IFOS B in Y such that f(p(, )) B, there exists an IFGSPROS A in X such that p(, ) A and f(A) B.

   Proof: (i) (ii) is obvious, since f -1(Ac) = (f -1(A))c.

   (ii) (iii) Let B be any IFOS in Y and let p(, ) X. Given f(p(, )) B. By hypothesis f -1(B) is an IFGSPROS in X.

   Proof: Let A be an IFCS in Y. Then f -1(V) is an IFRCS in

   X. Since every IFRCS is an IFGSPRCS, f -1(V) is an

   Take A= f -1(B). Now p

   (, )

   f -1(f(p

   (, )

   ). Therefore

   IFGSPRCS in X. Hence f is an IFGSPR continuous mapping.

   f -1(f(p(, )) f -1(B) = A. This implies p(, A and f(A) = f(f -1(B)) B.

   (iii) (i) Let A be an IFCS in Y. Then its complement, say B = Ac, is an IFOS in Y. Let p(, ) C and f(C) B. Now

   is an IFGSPROS in Y. Therefore f -1(spint(B)) is an IFGSPROS in X, by hypothesis. Since X is an IFSPRT1/2

   C f -1(f(C)) f -1(B). Thus p(, )

   -1

   f -1(B). Therefore

   -1 c

   space, f -1(spint(B)) is an IFSPOS in X. Hence

   f -1(spint(B)) = spint(f-1(spint(B))) spint(f -1(B)).

   f (B) is an IFGSPROS in X by Result 2.27. That is f (A )

   is an IFGSPROS in X and hence f -1(A) is an IFGSPRCS in

   X. Thus f is an IFGSPR continuous mapping.

4. Intuitionistic Fuzzy Generalized Semipre Regular Irresolute Mappings

In this section we have introduced intuitionistic fuzzy generalized semipre regular irresolute mappings and studied some of their properties.

Definition 4.1:

A mapping f: (X, ) (Y, ) be an intuitionistic fuzzy generalized semipre regular irresolute (IFGSPR irresolute) mapping if f -1(V) is an IFGSPRCS in (X, ) for every IFGSPRCS V of (Y, ).

Theorem 4.2:

Let f: (X, ) (Y, ) and g: (Y, ) (Z, ) be IFGSPR irresolute mapping. Then g f : (X, ) (Z, ) is an IFGSPR irresolute mapping.

Proof: Let V be an IFGSPRCS in Z. Then g -1(V) is an IFGSPRCS in Y. Since f is an IFGSPR irresolute,

f-1(g -1(V)) is an IFGSPRCS in X, by hypothesis. Hence g f is an IFGSPR irresolute mapping.

Theorem 4.3:

Let f: (X, ) (Y, ) be an IFGSPR irresolute mapping and

g: (Y, ) (Z, ) be IFGSPR continuous mapping, the g f : (X, ) (Z, ) is an IFGSPR continuous mapping.

Proof: Let V be an IFCS in Z. Then g -1(V) is an IFGSPRCS in Y. Since f is an IFGSPR irresolute mapping, f -1(g -1(V)) is an IFGSPRCS in X. Hence g f is an IFGSPR continuous mapping.

Theorem 4.4:

Let f: (X, ) (Y, ) be a mapping from an IFTS X into an IFTS Y. Then the following conditions are equivalent if X and Y are IFSPRT1/2 space:

1. f is an IFGSPR irresolute mapping,

2. f -1(B) is an IFGSPROS in X for each IFGSPROS B in Y,

3. f -1(spint(B)) spint(f -1(B)) for each IFS B of Y,

4. spcl(f -1(B)) f -1(spcl(B)) for each IFS B of Y.

Proof: (i) (ii) is obvious, since f -1(Ac) = (f -1(A))c.

(ii) (iii) Let B be any IFS in Y and spint(B) B. Also f -1(spint(B)) f -1(B). Since spint(B) is an IFSPOS in Y, it

1. (iv) is obvious by taking complement in (iii).

2. (i) Let B be an IFGSPRCS in Y. Since Y is an

IFSPRT1/2 space, B is an IFSPCS in Y and spcl(B) = B. Hence f -1(B) = f -1(spcl(B)) spcl(f -1(B)), by hypothesis. But f spcl(f -1(B)). Therefore spcl(f -1(B)) = f -1(B). This implies f -1(B) is an IFSPCS and hence it is an IFGSPRCS in X. Thus f is an IFGSPR irresolute mapping.

Theorem 4.5:

Let f : (X, ) (Y, ) be a an IFGSPR irresolute mapping from an IFTS X into an IFTS Y. Then f spint(f -1 (cl(int(cl(B))))) for every IFGSPROS B in Y, if X and Y are IFSPRT1/2 spaces.

Proof: Let B be an IFGSPROS in Y. Then by hypothesis f -1(B) is an IFGSPROS in X. Since X is an IFSPRT1/2 space, f -1(B) is an IFSPOS in X. Therefore spint(f -1(B)) = f -1(B). Since Y is an IFSPRT1/2 space, B is an IFSPOS in Y and B cl(int(cl(B))). Now f -1(B) = spint(f -1(B)) implies,

f -1(B) spint(f -1(cl(int(cl(B)))).

Theorem 4.6:

Let f: (X, ) (Y, ) be a an IFGSPR irresolute mapping from an IFTS X into an IFTS Y. Then f spint(cl(int(cl(f -1(B))))) for every IFGSPROS B in Y, if X and Y are IFSPRT1/2 spaces.

Proof: Let B be an IFGSPROS in Y. Then by hypothesis f -1(B) is an IFGSPROS in X. Since X is an IFSPRT1/2 space, f -1(B) is an IFSPOS in X. Threfore spint(f -1(B)) = f -1(B) cl(int(cl(f -1(B)))). Hence f -1(B)

spint(cl(int(cl(f -1(B))))).

Definition 4.7:

A function (X, ) is said to be a gsprIFTrg space if every intuitionistic fuzzy generalized semipre regular closed set is an intuitionistic fuzzy regular generalized closed set.

Theorem 4.8:

Let f: (X, ) (Y, ) be a mapping from an IFTS X into an IFTS Y. Then the following conditions are equivalent if X and Y are gsprIFTrg space:

1. f is an IFRG irresolute mapping,

2. f is an IFGSPR irresolute mapping.

   Proof: (i) (ii): Let f: (X, ) (Y, ) be an IFRG irresolute. Let V be an IFGSPRCS in Y. As Y is gsprIFTrg

   space, then V is an IFRGCS in Y. Since f is an IFRG irresolute, f-1(V) is an IFRGCS in X. But every IFRGCS is an IFGSPRCS in X and hence f-1(V) is an IFGSPRCS in X. Therefore, f is an IFGSPR irresolute mapping.

   (ii) (i): Let f: (X, ) (Y, ) be an IFGSPR irresolute. Let V be an IFRGCS in Y. But every IFRGCS is an IFGSPRCS and hence V is an IFGSPRCS in Y and f is an IFGSPR irresolute implies f-1(V) is an IFGSPRCS in X. But X is gsprIFTrg space and hence f-1(V) is an IFRGCS in X. Thus, f is an IFRG irresolute mapping.

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