Intuitionistic Fuzzy Almost Generalized (Beta) Closed Mappings

Intuitionistic Fuzzy Almost Generalized (Beta) Closed Mappings

D. Jayanthi

Department of Mathematics, Avinashilingam University, Coimbatore,

Tamil Nadu, India.

ABSTRACT:

In this paper we introduce intuitionistic fuzzy almost generalized closed mappings and intuitionistic fuzzy almost generalized open mappings. We investigate some of their properties. Also we provide the relation between intuitionistic fuzzy almost generalized closed mappings and other intuitionistic fuzzy closed mappings.

Key words and phrases : Intuitionistic fuzzy topology, intuitionistic fuzzy generalized T1/2 space, intuitionistic fuzzy almost generalized closed mappings.

1. Introduction

   The notion of intuitionistic fuzzy sets is introduced by Atanassov [1]. Using this notion, Coker [2] introduced the notion of intuitionistic fuzzy topological spaces.

   D. Jayanthi [4] introduced the notion of intuitionistic fuzzy generalized closed mappings and intuitionistic fuzzy generalized open mappings. In this paper we introduce intuitionistic fuzzy almost generalized closed mappings. We investigate some of its properties. Also we provide the relation between an intuitionistic fuzzy almost generalized closed mapping and other intuitionistic fuzzy closed mappings.

2. Preliminaries

   Definition 2.1: [1] An intuitionistic fuzzy set (IFS) A in X is an object having the form

   A = { x, A (x), A(x) / x X}

   where the functions A: X [0,1] and A: X [0,1] denote the degree of membership (namely A(x)) and the degree of non -membership (namely A(x)) of each element x X to the set A, respectively, and 0 A (x) + A(x) 1 for each x

   X.

   Definition 2.2: [1] Let A and B be IFSs of the form A = { x, A (x), A(x) / xX } and B = { x, B (x), B(x) / x X }.

   Then

   1. A B if and only if A (x) B (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), A(x) / x X }

   (d) A B = { x, A (x) B (x), A(x) B(x) / x X }

   (e) A B = { x, A (x) B (x), A(x) B(x) / x X }

   For the sake of simplicity, we shall use the notation A = x, A, A instead of A = {x, A (x), A(x)/xX}.

   The intuitionistic fuzzy sets 0~ = {x, 0, 1/xX} and 1~ = {x, 1, 0/xX} are respectively the empty set and the whole set of X.

   Definition 2.3: [2] Let , [0, 1] with + 1. An intuitionistic fuzzy point (IFP), written as p(, ), is defined to be an IFS of X given by

   p(, )(x) = (, ) if x = p,

   = (0, 1) otherwise.

   Definition 2.4: [2] An intuitionistic fuzzy topology (IFT) on X is a family of IFSs in X satisfying the following axioms.

   (i) 0~, 1~

   1. G 1 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) and any IFS in is known as an intuitionistic fuzzy open set (IFOS) in X. The complement Ac of an IFOS A in IFTS (X, ) is called an intuitionistic fuzzy closed set (IFCS) in X.

   Definition 2.5:[2] Let (X, ) be an IFTS and A = x, A, A be an IFS in X. Then the intuitionistic fuzzy interior and intuitionistic fuzzy closure are defined by

   int (A) = { G / G is an IFOS in X and G A } cl (A) = { K / K is an IFCS in X and A K }

   Definition 2.6:[3] An IFS A = x, A, A in an IFTS (X, ) is said to be an

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

   2. intuitionistic fuzzy semi closed set (IFSCS) if int(cl(A)) A

   3. intuitionistic fuzzy pre closed set (IFPCS) if cl(int(A)) A

   4. intuitionistic fuzzy closed set (IFCS) if cl(int(cl(A)) A

   5. intuitionistic fuzzy closed set (IFCS) if int(cl(int(A))) A

   The respective complements of the above IFCSs are called their respective IFOSs.

   Definition 2.7:[2] Let A be an IFS in an IFTS (X, ). Then the interior and the closure of A are defined by

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

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

   Definition 2.8:[5] An IFS A in an IFTS (X, ) is said to be an intuitionistic fuzzy generalized closed set (IFGCS) if cl(A) U whenever A U and U is an IFOS in (X, ).The complement Ac, is called an intuitionistic fuzzy generalized open set (IFGOS) in X.

   Definition 2.9:[3] Let p(, ) be an IFP of (X, ). An IFS A of X is called an intuitionistic fuzzy neighborhood (IFN) of p(, ) if there exists an IFOS B in X such that p(, ) B A.

   Definition 2.10:[5] If every IFGCS in (X, ) is an IFCS in (X, ), then the space can be called as an intuitionistic fuzzy T1/2 space (IFT1/2 space).

   Definition 2.11:[3] A map f: X Y is called an intuitionistic fuzzy closed mapping (IFCM) if f(A) is an IFCS in Y for each IFCS A in X.

   Definition 2.12:[3] A map f: X Y is called an

   1. intuitionistic fuzzy semi open mapping (IFSOM) if f(A) is an IFSOS in Y for each IFOS A in X.

   2. intuitionistic fuzzy open mapping (IFOM) if f(A) is an IFOS in Y for each IFOS A in X.

   3. intuitionistic fuzzy preopen mapping (IFPOM) if f(A) is an IFPOS in Y for each IFOS A in X.

   4. intuitionistic fuzzy open mapping (IFOM) if f(A) is an IFOS in Y for each IFOS A in X.

      Definition 2.13: [3] The intuitionistic fuzzy semi closure and the intuitionistic fuzzy

      closure of an IFS A in an IFTS (X, ) are defined by scl(A) = { K / K is an IFSCS in X and A K }.

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

      Definition 2.14:[5] Let p(, ) be an IFP in (X, ). An IFS A of X is called an intuitionistic fuzzy neighborhood (IFN) of p(, ) if there is an IFOS B in X such that p(, ) B A.

      Definition 2.15: [4]A mapping f : X Y is said to be an intuitionistic fuzzy generalized closed mapping (IFGCM) if f(A) is an IFGCS in Y for every IFCS A in X.

      Definition 2.16: [4]A mapping f : X Y is said to be an intuitionistic fuzzy M- generalized closed mapping (IFMGCM) if f(A) is an IFGCS in Y for every IFGCS A in X.

      Definition 2.17: [5] An IFS A is said to be an intuitionistic fuzzy dense (IFD) in another IFS B in an IFTS (X, ), if cl(A) = B.

      Definition 2.18: [6] A mapping f : X Y is said to be an intuitionistic fuzzy generalized continuous mapping (IFGcts.M) if f -1(A) is an IFGCS in X for every IFCS A in Y.

3. Intuitionistic fuzzy almost generalized closed mappings and intuitionistic fuzzy almost generalized open mappigs.

In this section we introduce intuitionistic fuzzy almost generalized closed mappings and intuitionistic fuzzy almost generalized open mappings. We study some of their properties

Definition 3.1: A map f: X Y is called an intuitionistic fuzzy almost generalized

closed mapping (IFaGCM) if f(A) is an IFGCS in Y for each IFRCS A in X.

Example 3.2: Let X = {a, b}, Y = {u, v} and G 1 = x, (0.5, 0.4), (0.5, 0.6),

G2 = y, (0.2, 0.3), (0.8, 0.7). 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 IFaGCM.

The relation between various types of intuitionistic fuzzy closedness is given in the following diagram.

IFCM IFGCM

. IFCM IFPCM IFaGCM IFMGCM

IFSCM

The reverse implications are not true in general in the above diagram. Ths can be seen from the following examples.

Example 3.3: In Example 3.2, f is an IFaGCM but not an IFCM, since G1c = x, (0.5, 0.6), (0.5, 0.4) is an IFCS in X but f(G1c) = y, (0.5, 0.6), (0.5, 0.4) is

not an IFCS in Y .

Example 3.4: Let X = {a, b}, Y = {u, v} and G 1 = x, (0.5, 0.4), (0.5, 0.6),

G2 = y, (0.5, 0.4), (0.2, 0.3). 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 IFaGCM but not an IFSCM, since G1c = x, (0.5, 0.6), (0.5, 0.4) is an IFCS in X but f(G1c) = y, (0.5, 0.6), (0.5, 0.4) is not an IFSCS in Y, since int(cl(f(G1c))) = 1~ f(G1c).

Example 3.5: In Example 3.4 f is an IFaGCM but not an IFCM, since G1c = x, (0.5, 0.6), (0.5, 0.4) is an IFCS in X but f(G1c) = y, (0.5, 0.6), (0.5, 0.4) is not an IFCS in Y, since cl(int(cl(f(G1c)))) = 1~ f(G1c).

1

1

Example 3.6: In Example 3.2 f is an IFaGCM but not an IFPCM, since G1c = x, (0.5, 0.6), (0.5, 0.4) is an IFCS in X but f(G c) = y, (0.5, 0.6), (0.5, 0.4) is not an IFPCS in Y, since cl(int(f(G1c))) = G2c f(G1c) .

Example 3.7: Let X = {a, b}, Y = {u, v} and G1 = x, (0.2, 0.2), (0.4, 0.4), G2 = x, (0.2, 0), (0.5, 0.4), G3 = y, (0.5, 0.6), (0.2, 0) and G4 = x, (0.4, 0.1), (0.2, 0.1),.

Then = {0~, G1, G2, 1~} and = {0~, G3, G4, 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 IFaGM but

not an IFGCM, since G2c = x, (0.5, 0.4), (0.2, 0) is an IFCS in X but f(G2c) = y, (0.5, 0.4), (0.2, 0) is not an IFGCS in Y, since f(G2c) G3 but cl(f(G2c) = 1~ G3.

Example 3.8: In Example 3.7 f is an IFaGCM but not an IFMGCM, since A = x, (0.4, 0.2), (0.2, 0) is an IFGCS in X but f(A) = y, (0.4, 0.2), (0.2, 0) is not an IFGCS in Y, since f(A) G3 but cl(f(A)) = 1~ G3.

Definition 3.9: A map f: X Y is called an intuitionistic fuzzy almost generalized

open mapping (IFaGOM) if f(A) is an IFGOS in Y for each IFROS A in X. An IFaGOM is an IFaGCM if it is a bijective mpping.

Theorem 3.10: Let p(, ) be an IFP in X. A bijective mapping f: X Y is an IFaGCM if for every IFOS A in X with f -1(p(, )) A, there exists an IFOS B in Y with p(, ) B such that f (A) is IFD in B.

Proof: Let A be an IFROS in X. Then A is an IFOS in X. Let f -1(p(, )) A, then

there exists an IFOS B in Y such that p(, ) B and cl(f (A)) = B. Since B is an IFOS, cl(f(A)) = B is also an IFOS in Y. Therefore int(cl(f(A))) = cl(f (A)). Now f

1. cl(f (A)) = int(cl(f(A))) cl(int(cl(f(A)))). This implies f(A) is an IFOS in Y and hence an IFGOS in Y. Thus f is an IFaGCM.

   Theorem 3.11: Let f: X Y be a bijective mapping where Y is an IFT1/2 space.

   Then the following are equivalent

   1. f is an IFaGCM.

   2. cl(f (A)) f (cl(A)) for every IFOS A in X

   3. cl(f (A)) f (cl(A)) for every IFSOS A in X

   4. f (A) int(f (int(cl(A)))) for every IFPOS A in X.

Proof: (i) (ii) Let A be an IFOS in X. Then cl(A) is an IFRCS in X. By hypothesis f(A) is an IFGCS in Y and hence is an IFCS in Y, since Y is an IFT1/2 space. This implies cl(f(cl(A))) = f (cl(A)). Now cl(f(A)) cl(f(cl(A))) = f(cl(A)). Thus cl(f(A)) f(cl(A)).

1. (iii) Since every IFSOS is an IFOS, the proof directly follows.

2. (i) Let A be an IFRCS in X. Then A = cl(int(A)). Therefore A is an IFSOS in

X. By hypothesis, cl(f(A)) f(cl(A)) = f(A) cl(f(A)). Hence f(A) is an IFCS and hence is an IFGCS in Y. Thus f is an IFaGCM.

(i) (iv) Let A be an IFPOS in X. Then A int(cl(A)). Since int(cl(A)) is an IFROS in X, by hypothesis, f(int(cl(A))) is an IFGOS in Y. Since Y is an IFT1/2 space, f(int(cl(A))) is an IFOS in Y. Therefore f(A) f(int(cl(A)))

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

1. (i) Let A be an IFROS in X. Then A is an IFPOS in X. By hypothesis, f(A) int(f(int(cl(A)))) = int(f(A)) f(A). This implies f(A) is an IFOS in Y and hence is an IFGOS in Y. Therefore f is an IFaGCM.

   Theorem 3.12: Let f: X Y be a map. Then f is an IFaGCM if for each IFP p(, )

   Y and for each IFOS B in X such that f -1(p(, )) B , cl(f(B)) is an IFN of p(,

   ) Y.

   Proof: Let p(, ) Y and let A be an IFROS in X. Then A is an IFOS in X. By hypothesis f -1(p(, )) A, that is p(, ) f(A) in Y and cl((f(A)) is an IFN of p(, )

   in Y. Therefore there exists an IFOS B in Y such that p(, )) B cl(f(A)). We have p(, ) f(A) cl(f(A)). Now B = { p(, ) / p(, ) B} = f (A). Therefore f(A) is an IFOS in Y and hence an IFGOS in Y. Thus f is an IFaGOM. By Theorem f is an IFaGCM.

   Theorem 3.13: Let f: X Y be a mapping where Y is an IFT1/2 space. Then the following are equivalent.

   1. f is an IFaGOM

   2. for each IFP p(, ) in Y and each IFROS B in X such that f -1 (p(, )) B, cl(f(cl(B))) is an IFN of p(, ) in Y.

Proof: (i) (ii) Let p(, ) Y and let B be an IFROS in X such that f -1(p(, )) B.

That is p(, ) f(B). By hypothesis, f(B) is an IFGOS in Y. Since Y is an IFT1/2 space, f(B) is an IFOS in Y. Now p(, ) f(B) f(cl(B)) cl(f(cl(B))). Hence cl(f(cl(B))) is an IFN of p(, ) in Y.

1. (i) Let B be an IFOS in X. Then f -1(p(, )) B. This implies p(, ) f(B). By

   hypothesis, cl(f(cl(B))) is an IFN of p(, ). Therefore there exists an IFOS A in Y such that p(, ) A cl(f(cl(B))). Now A = { p(, ) / p(, ) A} = f(B). Therefore f(B) is an IFOS and hence an IFGOS in Y. Thus f is an IFaGOM.

   Theorem 3.14: The following are equivalent for a mapping f: X Y where y is an IFT1/2 space.

   1. f is an IFaGCM

   2. cl(f(A)) f(cl(A)) for every IFOS A in X

   3. cl(f(A)) f(cl(A)) for every IFSOS A in X

   4. f(A) int(f(scl(A))) for every IFPOS A in X.

Proof: (i) (ii) Let A be an IFOS in X. Then cl(A) is an IFRCS in X. By hypothesis f(A) is an IFGCS in Y and hence is an IFCS in Y, since Y is an IFT1/2 space. This implies cl(f(cl(A))) = f(cl(A)). Now cl(f(A)) cl(f(cl(A))) = f(cl(A)). Since cl(A) is an IFRCS, cl(int(cl(A))) = cl(A). Therefore cl(f(A)) f(cl(A)) = f(cl(int(cl(A)))) f(A cl(int(cl(A)))) f(cl(A)). Hence cl(f(A)) f(cl(A)).

1. (iii) Let A be an IFSOS in X. Since every IFSOS is an IFOS, the proof is obvious.

2. (i) Let A be an IFRCS in X. Then A = cl(int(A)). Therefore A is an IFSOS in

X. By hypothesis, cl(f(A)) f(cl(A)) f(cl(A)) = f(A) cl(f(A)). That is

cl(f(A)) = f(A). Hence f(A) is an IFCS and hence is an IFGCS in Y. Thus f is an IFaGCM.

(i) (iv) Let A be an IFPOS in X. Then A int(cl(A)). Since int(cl(A)) is an IFROS in X, by hypothesis, f(int(cl(A))) is an IFGOS in Y. Since Y is an IFT1/2 space, f(int(cl(A))) is an IFOS in Y. Therefore f(A) f(int(cl(A))) int(f(int(cl(A)))) =

int(f(A int(cl(A)))) = int(f(scl(A))). That is f(A) int(f(scl(A))).

1. (i) Let A be an IFROS in X. Then A is an IFPOS in X. By hypothesis, f(A)

   int(f(scl(A))). This implies f(A) int(f(A int(cl(A)))) int(f(A A)) =

   int(f(A )) f(A). Therefore f(A) is an IFOS in Y and hence an IFGOS in Y. Thus f is an IFaGCM.

   Next we provide the characterization theorem for an IFaGOM.

   Theorem 3.15: Let f: X Y be a bijective mapping. Then the following are equivalent.

   1. f is an IFaGOM

   2. f is an IFaGCM

   3. f -1 is an IFaGCts.M.

Proof: (i) (ii) is obvious.

1. (iii) Let A X be an IFRCS. Then by hypothesis, f(A) is an IFGCS in Y. That is (f -1) -1(A) is an IFGCS in Y. This implies f -1 is an IFaGCts.M.

2. (ii) Let A X be an IFRCS. Then by hypothess (f -1) -1(A) is an IFGCS in Y. That is f(A) is an IFGCS in Y. Hence f is an IFaGCM.

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(submitted)