Fuzzy Supra Regular Open Sets

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Fuzzy Supra Regular Open Sets

J.Srikiruthika

#1

, Dr.A.Kalaichelvi

*2

#1

Assistant Professor, Faculty of Engineering, Department of Science and Humanities, Avinashilingam Institute for Home Science and

Higher Education for Women, Coimbatore – 641 043, Tamil Nadu, India.

1

[email protected]

*2

Professor, Department of Mathematics, Avinashilingam Institute for Home Science and

Higher Education for Women, Coimbatore – 641 043, Tamil Nadu, India.

2

[email protected]

Abstract— In this paper fuzzy supra regular open sets and fuzzy supra regular closed sets are introduced and certain properties and relations of fuzzy supra regular open sets and fuzzy supra regular closed sets are investigated.

Keywords— Fuzzy supra topological space, fuzzy supra regular open set, fuzzy supra regular closed set.

1. INTRODUCTION

In 1937, Stone [9] introduced the notion of regular open sets in topological spaces. In 1981, Azad [3] introduced the concept of fuzzy regular open sets in fuzzy topological spaces.

In 1983, Mashhour et al. [6] introduced the concept of supra topological spaces. In 1987, Abd El-Monsef et al. [1] introduced fuzzy supra topological spaces. In 2016, El-Shafei et al. [5] introduced supra regular open sets. In this paper the concept of fuzzy supra regular open sets and fuzzy supra regular closed sets are introduced and studied.

2. PRELIMINARY DEFINITIONS

Throughout the paper X denotes a non-empty set.

Definition: 2.1 [10]

A fuzzy set in X is a map f: X → I = [0,1] . The family of fuzzy sets of X is denoted by IX.

Following are some basic operations of fuzzy sets in X. For the fuzzy sets f and g in X, 1) f = g if and only if f(x) = g(x) for all x ϵ X 2) f ≤ g if and only if f(x) ≤ g(x) for all x ϵ X 3) (f ᴠ g )(x) = max { f(x),g(x) } for all x ϵ X 4) (f ᴧ g )(x) = min { f(x),g(x) } for all x ϵ X 5) f c(x) = 1- f(x) for all x ϵ X here fc denotes

the complement of f. 6) f ≤ g if and only if f c ≥ g c

7) For a family { f λ/ λ ϵ ᴧ } of fuzzy sets defined on a set X

) (x) = (x) =

8) For any α ϵ I, the constant fuzzy set α in X is a fuzzy set in X defined by α(x) = α for all x ϵ X.

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0 denotes null fuzzy set in X and 1 denotes universal

fuzzy set in X. Definition: 2.2 [4]

A fuzzy topological space is a pair (X, δ) where X is a nonempty set and δ is a family of fuzzy set on X satisfying the following properties:

1) The constant functions 0 and 1 belong to δ. 2) f, g ϵ δ implies f ᴧ g ϵ δ.

3) fλ ϵ δfor each λ ϵ ᴧ implies

Then δ is called a fuzzy topology on X. Every member of δ is called fuzzy open set. The complement of a fuzzy open set is called fuzzy closed set.

The closure and interior of a fuzzy set f ϵ IX are defined respectively as

cl (f) = ᴧ { g / g is a fuzzy closed set in X and f ≤ g } int (f)= ᴠ { g / g is a fuzzy open set in X and g ≤ f } Clearly cl(f) is the smallest fuzzy closed set containing f and int(f) is the largest fuzzy open set contained in f.

Remark: 2.4[3]

For a family {fλ / λ ϵ ᴧ } of fuzzy sets in a fuzzy topological space (X, δ)

1. ᴠ cl( f λ ) ≤ cl( ᴠ f λ ) 2. ᴠ int ( f λ ) ≤ int ( ᴠ f λ ) If the collection is finite, then

ᴠ cl( f λ ) = cl( ᴠ f λ )

A collection δ* of fuzzy sets in a set X is called fuzzy supra topology on X if the following conditions are satisfied:

1) 0 and 1 belongs to δ*.

2) f λϵ δ*for each λ ϵ ᴧ implies . The pair (X, δ*) is called a fuzzy supra topological space. The elements of δ*are called fuzzy supra open sets and the complement of a fuzzy supra open set is called fuzzy supra closed set.

Let (X, δ*) be a fuzzy supra topological space and f be a fuzzy set in X, then the fuzzy supra closure and fuzzy supra interior of f defined respectively as

cl*(f) = ᴧ { g / g is a fuzzy supra closed set in X and f ≤ g}

int *(f) = ᴠ { g / g is a fuzzy supra open set in X and g ≤ f}

Let (X, δ) be a fuzzy topological space and δ* be a fuzzy supra topology on X .We call δ* a fuzzy supra topology associated with δ if δ ≤ δ*

Remark: 2.8 [8]

1) The fuzzy supra closure of a fuzzy set f in a fuzzy supra topological space is the smallest fuzzy supra closed set containing f.

2) The fuzzy supra interior of a fuzzy set f in a fuzzy supra topological space is the largest fuzzy supra open set contained in f.

3) If (X, δ*) is an associated fuzzy supra topological space with the fuzzy topological space (X, δ) and f is any fuzzy set in X, then int (f) ≤ int *(f) ≤ f ≤ cl *(f) ≤ cl (f) Theorem: 2.9 [8]

For any two fuzzy sets f and g in a fuzzy supra topological space (X, δ*),

1) f is a fuzzy supra closed set if and only if cl *(f) = f.

2) f is a fuzzy supra open set if and only if int*(f) = (f).

3) f ≤ g implies int*(f) ≤ int *(g) and cl*(f) ≤ cl *

(g)

4) cl*( cl*(f)) = cl*(f) and int*(int*(f)) = int*(f). 5) cl*(f ᴠ g) ≥ cl*(f) ᴠ cl *(g)

6) cl*(f ᴧ g) ≤ cl*

(f) ᴧ cl*(g) 7) int*(f ᴠ g) ≥ int*(f) ᴠ int *(g) 8) int*(f ᴧ g) ≤ int*(f) ᴧ int *(g) 9) cl*(f c) = [int*(f)] c

10) int*(f c) = [cl*(f)] c

3. FUZZY SUPRA REGULAR OPEN SETS Definition: 3.1

Let (X, δ*) be a fuzzy supra topological space. A fuzzy set f is called a fuzzy supra regular open set if

int *(cl*(f)) = f

and a fuzzy set f is called a fuzzy supra regular closed set if

cl *(int*(f)) = f

International Journal of Pure and Applied Mathematics Special Issue

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Theorem: 3.2

Let f be a fuzzy set in a fuzzy supra topological space (X, δ*),then f is a fuzzy supra regular open set if and only if f c is a fuzzy supra regular closed set.

Proof:

Let f be a fuzzy supra regular open set int *(cl*(f )) = f

(int *(cl*(f )))c = f c cl* ((cl*(f ))c) = f c cl*(int*(f c )) = f c

f c is a fuzzy supra regular closed set Conversely

Let f c be a fuzzy supra regular closed set cl*(int*(f c )) = f c

(cl*(int*(f c )))c = (f c )c int*((int*(f c ))c) = f int*(cl*((f c )c)) = f int*(cl*((f )) =f

f c is a fuzzy supra regular open set

Theorem: 3.3

Every fuzzy supra regular open set is fuzzy supra open set.

Proof:

Let f be a fuzzy supra regular open set in (X, δ*) Then int*(cl*(f)) = f

By the definition of fuzzy supra interior, int*(f) is the largest fuzzy supra open set contained in f.

int*(cl*(f)) is the largest fuzzy supra open set contained in cl*(f)

f is a fuzzy supra open set. Theorem: 3.4

Every fuzzy supra regular closed set is fuzzy supra closed set.

Proof:

Let f be a fuzzy supra regular closed set in (X, δ*) Then cl*(int*(f)) = f

By the definition of fuzzy supra closure, cl*(f) is the smallest fuzzy supra closed set containing f. cl *(int*(f)) is the smallest fuzzy supra closed set containing int*(f)

f is a fuzzy supra closed set. Theorem: 3.5

(1) The fuzzy supra closure of a fuzzy supra open set is a fuzzy supra regular closed set. (2) The fuzzy supra interior of a fuzzy supra

closed set is a fuzzy supra regular open set. Proof:

(1) Let f be a fuzzy supra open set in a fuzzy supra topological space (X, δ*).

By the definition of fuzzy supra interior, int*(f)) ≤ f int *(cl*(f)) ≤ cl*(f) cl*(int *(cl*(f))) ≤ cl*(cl*(f)) cl*(int *(cl*(f))) ≤ cl*(f) ---(1) Since f ≤ cl*(f) int*(f) ≤ int *(cl*(f)) f ≤ int *(cl* (f)) cl*(f) ≤ cl*(int *(cl* (f)))---(2) From (1) and (2) cl*(int *(cl*(f))) = cl*_(f) fuzzy supra closure of a fuzzy supra open set is a fuzzy supra regular closed set.

(2) Let f be a fuzzy supra closed set in a fuzzy supra topological space (X, δ*).

By the definition of fuzzy supra closure, f ≤ cl*(f)) int*(f)) ≤ cl*(int *(f))

int*(int*(f)) ≤ int*(cl*(int*(f)))

int*(f) ≤ int*(cl*(int*(f)))---(3) Since int*(f) ≤ f

cl*(int*(f)) ≤ cl*(f) cl*(int*(f)) ≤ f

int*(cl*(int* (f))) ≤ int*(f) ---(4) From (3) and (4)

int*(cl*(int* (f))) = int*(f)

fuzzy supra interior of a fuzzy supra closed set is a fuzzy supra regular open set.

Definition: 3.6

Let (X, δ*) be a fuzzy supra topological space and f be a fuzzy set in X, then the fuzzy supra regular closure and fuzzy supra regular interior of f is denoted and defined respectively as

rcl*(f) = ᴧ { g / g is a fuzzy supra regular closed set in X and f ≤ g}

rint *(f) = ᴠ { g / g is a fuzzy supra regular open set in X and g ≤ f}

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Remark: 3.7

It is obvious that rint*(f) is a fuzzy supra regular open set and rcl*(f) is a fuzzy supra regular closed set.

Theorem: 3.8

For any fuzzy set f in a fuzzy supra topological space (X,δ*),

1) [rint *(f)]c = rcl *(f c) 2) [rcl *(f)]c = rint *(f c) Proof:

1) consider

rint*(f) = ᴠ { g / g is a fuzzy supra regular open set in X and g ≤ f} [rint*(f)]c = 1 - ᴠ { g / g is a fuzzy supra

regular

open set in X and g ≤ f} = ᴧ { gc

/ gc is a fuzzy supra regular

closed set in X and gc ≥ f c} = rcl*(f c)

2)

rcl*(f) = ᴧ { g / g is a fuzzy supra regular closed set in X and f ≤ g}

[rcl*(f)]c = 1 - ᴧ { g / g is a fuzzy supra regular

closed set in X and f ≤ g}

= ᴠ { gc

/ gc is a fuzzy supra regular open set in X and f c ≥ gc

} = rint*(f c)

Theorem: 3.9

For any two fuzzy sets f and g in fuzzy supra topological space (X,δ*),then if f ≤ g implies 1) rint*(f ) ≤ rint*(g )

2) rcl*(f ) ≤ rcl*(g ) Proof:

1) rint*(f) = ᴠ { h / h is a fuzzy supra regular open set in X and h ≤ f}

Since rint*(f) is the largest fuzzy supra regular open set contained in f

rint*(f) ≤ f f ≤ g implies rint*(f) ≤ f ≤ g implies rint*(f) ≤ g

rint*(f) is the fuzzy supra regular open set

contained in g.

But rint*(g) is the largest fuzzy supra regular open set contained in g.

rint*(f) ≤ rint*(g) 2) f ≤ g if and only if f c ≥ g c consider g c ≤ f c rint*(gc ) ≤ rint*(f c) (rcl *(g) )c ≤ (rcl *(f ))c rcl *(f ) ≤ rcl *(g ) REFERENC ES

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[2] Anuradha.N and Baby Chacko., “ On Supra R-open sets and Supra r-continuity”, journal of Computer and Mathematical Science, Vol 7(8),(August 2016), 412 -419.

[3] S.Abinaya 2G.Arulkumaran, “Detecting Black Hole Attack Using Fuzzy T rust Approach in MANET ”, International Journal of Innovations in Scientific and Engineering Research (IJISER), Vol.4, No.3, pp.102 -108, 2017. [4] Azad. K.K., “ On fuzzy semi continuity, fuzzy almost

continuity and fuzzy weakly continuity”, Journal of Mathematical Analysis and Applications 82, 14-32, (1981). [5] Chang C.L., “ Fuzzy topological spaces”, Journal of Mathematical Analysis and Application, Vol. 24, pp.182-190, (1968).

[6] El-Shafei, M.E.,Abo-elhamayel, M., and Al-Shami T .M., “On Supra R-open sets and some applications on topological spaces”, Journal of progressive research in Mathematics(JPRM), Vol 8, Issue 2, (July 2016), 1237 -1248.

[7] MAI Navid, NH Niloy, “Data Mining Algorithm on Fuzzy Weighted Association Rules”, International Research Journal of Multidisciplinary Science & Technology (irjmrs), Vol.1, No.7, pp.480-488, 2016.

[8] Mashhour A. S., Allam A. A., Mahmoud F. S and Khedr F. H., “ On supra topological spaces”, Indian J. Pure and Appl. Math. no.4, 14, pp.502-510, (1983).

[9] Njastad .O. “ On some classes of nearly open sets”, Pacific J. Math. 15, pp.961-970(1965).

[10] Sahidul Ahmed and Biman Chandra Chetia, “ On Certain Properties of Fuzzy Supra Sem i open Sets”, International Journal of Fuzzy Mathematics and Systems, Vol. 4, No 1,pp. 93-98,(2014).

[11] Stone M., “ Application of the theory of Boolean rings to general topology”, Trans. Amer. Math. Soc., 41, (1937), 374–481.

[12] Zadeh L.A., “ Fuzzy sets”, Inform. and Control, Vol.8, pp.338-353, (1965).

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