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A Novel on NSR Contra Strong Precontinuity.

1 Introduction

L. A. Zadeh introduced the idea of fuzzy sets in 1965[16] and later Atanassov [1] generalized it and offered the concept of intuitionistic fuzzy sets. Intuitionistic fuzzy set theory has applications in many fields like medical diagnosis, information technology, nanorobotics, etc. The idea of intutionistic L-fuzzy subring was introduced by K. Meena and V. Thomas [9]. R. Narmada Devi et al. [10, 11, 12] introduced the concept of contra strong precontinuity with respect to the intuitionistic fuzzy structure ring spaces and B. Krteska and E. Ekici [5, 7, 8] introduced the idea of intuitionistic fuzzy contra continuity. The concept of [alpha] continuity in intuitionistic fuzzy topological spaces was introduced by J. K. Jeon et al. [6]. F. Smarandache introduced the important and useful concepts of neutrosophy and neutrosophic set [[14], [15]]. A. A. Salama and S. A. Alblowi were established the concepts of neutrosophic crisp set and neutrosophic crisp topological space[13]. In this paper, the concept of NSR contra continuous function is introduced. Several types of contra-continuous functions in NSR spaces are discussed. Some interesting properties of NSR contra strongly precontinuous function is established.

2 Preliminiaries

Definition 2.1. [14, 15] Let T, I, F be real standard or non standard subsets of][0.sup.-], [1.sup.+] [, with

(i) [sup.sub.T] = [t.sub.sup], in[f.sub.T] = [t.sub.in f]

(ii) [sup.sub.I] = [i.sub.sup], in [f.sub.i] = [i.sub.in f]

(iii) [sup.sub.F] = [f.sub.sup], in [f.sub.F] = [f.sub.in f]

(iv) n - sup = [t.sub.sup] + [i.sub.sup] + [f.sub.sup]

(v) n - in f = [t.sub.in f] + [i.sub.in f] + [f.sub.in f].

Observe that T, I, F are neutrosophic components.

Definition 2.2. [14, 15]Let [S.sub.1] be a non-empty fixed set. A neutrosophic set (briefly N-set) [LAMBDA] is an object such that [LAMBDA] = {<u, [[mu].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u), [[gamma].sub.[LAMBDA]] (u)> : u [member of] [S.sub.1]} where [[mu].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u) and [[gamma].sub.[LAMBDA]] (u) which represents the degree of membership function (namely [[mu].sub.[LAMBDA]] (u)), the degree of indeterminacy (namely [[sigma].sub.[LAMBDA]] (u)) and the degree of non-membership (namely [[gamma].sub.[LAMBDA]] (u)) respectively of each element u [member of] [S.sub.1] to the set [LAMBDA].

Definition 2.3. [13] Let [S.sub.1] [not equal to] [empty set] and the N-sets [LAMBDA] and [GAMMA] be defined as

[LAMBDA] = {<u, [[mu].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u), [[GAMMA].sub.[LAMBDA]] (u)> : u [member of] [S.sub.1]}, [GAMMA] = {<u, [[mu].sub.[GAMMA]] (u), [[sigma].sub.[GAMMA]] (u), [[GAMMA].sub.[GAMMA]] (u)> : u [member of] [S.sub.1]}. Then

(a) [LAMBDA] [??] [GAMMA] iff [[mu].sub.[LAMBDA]] (u) [less than or equal to] [[mu].sub.[GAMMA]] (u), [[sigma].sub.[LAMBDA]] (u) [less than or equal to] [[sigma].sub.[GAMMA]] (u) and [[GAMMA].sub.[LAMBDA]] (u) [greater than or equal to] [[GAMMA].sub.[GAMMA]] (u) for all u [member of] [S.sub.1];

(b) [LAMBDA] = [GAMMA] iff [LAMBDA] [??] [GAMMA] and [GAMMA] [??] [LAMBDA];

(c) [bar.[LAMBDA]] = {<u, [[GAMMA].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u), [[mu].sub.[LAMBDA]] (u)> : u [member of] [S.sub.1]}; [Complement of [LAMBDA]]

(d) [LAMBDA] [intersection] [GAMMA] = {<u, [[mu].sub.[LAMBDA]] (u) [and] [[mu].sub.[GAMMA]] (u), [[sigma].sub.[LAMBDA]] (u) [and] [[sigma].sub.[GAMMA]] (u), [[GAMMA].sub.[LAMBDA]] (u) [disjunction] [[GAMMA].sub.[GAMMA]] (u)> : u [member of] [S.sub.1]};

(e) [LAMBDA] [union] [GAMMA] = {<u, [[mu].sub.[LAMBDA]] (u) [disjunction] [[mu].sub.[GAMMA]] (u), [[sigma].sub.[LAMBDA]] (u) [disjunction] [[sigma].sub.[GAMMA]] (u), [[GAMMA].sub.[LAMBDA]] (u) [and] [[gamma].sub.[GAMMA]] (u)> : u [member of] [S.sub.1]};

(f) [][LAMBDA] = {<u, [[mu].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u), 1 - [[mu].sub.[LAMBDA]] (u)>> : u [member of] [S.sub.1]};

(g) <>[LAMBDA] = {<u, 1 - [[GAMMA].sub.[LAMBDA]] (u), [[sigma].sub.[LAMBDA]] (u), [[GAMMA].sub.[LAMBDA]](u)> : u [member of] [S.sub.1]}.

Definition 2.4. [13] Let {[[LAMBDA].sub.I] : i [member of] J} be an arbitrary family of N-sets in [S.sub.1]. Then

(a) [mathematical expression not reproducible];

(b) [mathematical expression not reproducible].

Definition 2.5. [13] [0.sub.N] = {<u, 0, 0, 1> : u [member of] S} and [1.sub.N] = {<u, 1, 1, 0> : u [member of] S}.

Definition 2.6. [4] A neutrosophic topology (briefly N-topology) on [S.sub.1] [not equal to] V is a family [[xi].sub.1] of N-sets in [S.sub.1] satisfying the following axioms:

(i) [0.sub.N], [1.sub.N] [member of] [[xi].sub.1],

(ii) [H.sub.1] [intersection] [H.sub.2] [member of] [[xi].sub.1] for any [H.sub.1], [H.sub.2] [member of] [[xi].sub.1],

(iii) [union] [H.sub.i] [member of] [[xi].sub.1] for arbitrary family {[G.sub.i] | i [member of] [LAMBDA]} [??] [[xi].sub.1].

In this case the ordered pair ([S.sub.1], [[xi].sub.1]) or simply [S.sub.1] is called an NTS and each N-set in [[xi].sub.1] is called a neutrosophic open set (briefly N-open set). The complement [bar.[LAMBDA]] of an N-open set [LAMBDA] in [S.sub.1] is called a neutrosophic closed set (briefly N-closed set) in [S.sub.1].

Definition 2.7. [13] Let D be any neutrosophic set in an neutrosophic topological space S. Then the neutrosophic interior and neutrosophic closure of D are defined and denoted by

(i) Nint(D) = [union] {H | H is an NS open set in Sand H [??] D}.

(ii) Ncl(D) = [intersection] {H | H is a neutrosophic closed set in S and H [??] D}.

Proposition 2.1. [13] For any neutrosophic set D in (S, [TAU]) we have

Ncl(C(D)) = C(Nint(D)) and Nint(C(D)) = C(Ncl(D)).

Corollary 2.1. [4] Let D, [D.sub.i](i [member of] J) and U, [U.sub.j](j [member of] K) IFSs in be [S.sub.1] and [S.sub.2] and [phi] : [S.sub.1] [right arrow] [S.sub.2] a function. Then

(i) D [??] [[phi].sup.-1]([phi](D)) (If [phi] is injective,then D = [[phi].sup.-1]([phi](D))),

(ii) [phi]([[phi].sup.-1](U)) [??] U (If [phi] is surjective,then [phi]([[phi].sup.-1](D)) = D),

(iii) [[phi].sup.-1]([union] [U.sub.j]) = [union] [[phi].sup.-1] ([U.sub.j]) and [[phi].sup.-1]([intersection][U.sub.j]) = [intersection] [[phi].sup.-1] ([U.sub.j]),

(iv) [[phi].sup.-1]([1.sub.~]) = [1.sub.~] and [[phi].sup.-1]([0.sub.~]) = [0.sub.~],

(v) [[phi].sup.-1]([bar.U])= [bar.[[phi].sup.-1](U)].

Definition 2.8. [5]

An IFS D of an IFTS is called an intuitionistic fuzzy [alpha]-open set (IF[alpha]OS) if D [??] int(cl(int(D))). The complement of an IF[alpha]OSis called an intuitionistic fuzzy [alpha]-closed set(IF[alpha]CS).

Definition 2.9. [3] A [phi] : X [right arrow] Y be a function.

(i) If B = {<v, [[mu].sub.B] (v), [[gamma].sub.B] (v)> : v [member of] Y} is an IFS in Y, then the preimage of B under [phi] (denoted by [[phi].sup.-1](B)) is defined by [[phi].sup.-1](B) = {<u, [[phi].sup.-1] ([[mu].sub.B]) (u),[[phi].sup.-1]([[gamma].sub.B]) (u)> : u [member of] X}.

(ii) If A = {<u, [[LAMBDA].sub.A] (U), [I.sub.A] (U)> : u [member of] X} is an IFS in X, then the image of A under [phi] (denoted by [phi](A)) is defined by [phi](A) = {<v, [phi]([[lambda].sub.A] (v)), (1 - [phi](1 - [I.sub.A]))(v)> : v [member of] Y}.

Definition 2.10. [3] Let (X, [tau]) and (Y, [sigma]) be two IFTSs and [phi] : X [right arrow] Y be a function. Then [phi] is said to be intuitionistic fuzzy continuous if the preimage of each IFS in [sigma] is an IFS in [tau].

Definition 2.11. [8, 9] Let R be a ring. An intuitionistic fuzzy set A = <u, [[mu].sub.A], [[gamma].sub.A]) in R is called an intuitionistic fuzzy ring on R if it satisfies the following conditions:

(i) [[mu].sub.A](u + v) [greater than or equal to] [[mu].sub.A](u) [and] [[mu].sub.A] (v) and [[mu].sub.A] (uv) [greater than or equal to] [[mu].sub.A] (u) [and] [[mu].sub.A] (v).

(ii) [[gamma].sub.A] (u + v) [less than or equal to] [[gamma].sub.A](u) [disjunction] [[gamma].sub.A](v) and [[gamma].sub.A] (uv) [less than or equal to] [[mu].sub.A](u) [disjunction] [[gamma].sub.A] (v).

for all u, v [member of] R.

3 Neutrosophic structure ring contra strong precontinuous function

In this section, the concepts of neutrosophic ring, neutrosophic structure ring space are introduced. Also some interesting properties of neutrosophic structure ring contra strong precontinuous function and their characterizations are studied.

Definition 3.1. Let R be a ring. A neutrosophic set [LAMBDA] = {<u, [[mu].sub.[LAMBDA](u)], [[sigma].sub.[LAMBDA](u)], [[gamma].sub.[LAMBDA](u)]> : u [member of] R} in R is called a neutrosophic ring[briefly NR] on R if it satisfies the following conditions:

(i) [[mu].sub.[LAMBDA](u+v)] [greater than or equal to] [[mu].sub.[LAMBDA](u)] [and] [[mu].sub.[LAMBDA](u)] and [[mu].sub.[LAMBDA](uv)] [greater than or equal to] [[mu].sub.[LAMBDA](u)] [and] [[mu].sub.[LAMBDA](v)].

(ii) [[sigma].sub.[LAMBDA](u+v)] [greater than or equal to] [[sigma].sub.[LAMBDA](u)] [and] [[sigma].sub.[LAMBDA](v)] and [[sigma].sub.[LAMBDA](uv)] [greater than or equal to] [[sigma].sub.[LAMBDA](u)] [and] [[sigma].sub.[LAMBDA](v)].

(iii) [[gamma].sub.[LAMBDA](u+v)] [less than or equal to] [[gamma].sub.[LAMBDA](u)] [disjunction] [[gamma].sub.[LAMBDA](v)] and [[gamma].sub.[LAMBDA](uv)] [less than or equal to] [[gamma].sub.[LAMBDA](x)] [disjunction] [[gamma].sub.[LAMBDA](y)].

for all u, v [member of] R.

Definition 3.2. Let R be a ring. A family S of a NR's in R is said to be neutrosophic structure ring on R if it satisfies the following axioms:

(i) [0.sub.N], [1.sub.N] [member of] S.

(ii) [H.sub.1] [intersection] [H.sub.2] [member of] S for any [H.sub.1], [H.sub.2] [member of] S.

(iii) [union] [H.sub.k] [member of] S for arbitrary family {[H.sub.k] | k [member of] J} [??] S.

The ordered pair (R, S) is called a neutrosophic structure ring(NSR) space. Every member of S is called a K open ring (briefly NOR) in (R, S). The complement of a NOR in (R, S) is a N closed ring (NCR)in (R, S).

Definition 3.3. Let D be a K ring in NSR space (R, S). Then NSR interior and NSR closure of D are defined and denoted by

(i) [Nint.sub.R](D) = [union]{H | H is a NOR in R and H [??] D}.

(ii) [Ncl.sub.R](D) = [intersection]{H | H is a NCR in R and H [??] D}.

Proposition 3.1. For any NR D in (R, S) we have

(i) [Ncl.sub.R](C(D)) = C([Nint.sub.R](D))

(ii) [Nint.sub.R](C(D)) = C([Ncl.sub.R](D))

Definition 3.4. A NR D of a NSR space (R, S) is said be a

(i) K regular open structure ring (NRegOSR), if D = [Nint.sub.R]([Ncl.sub.R](D))

(ii) N[alpha]-open structure ring (K[alpha]OSR), if D [??] [Nint.sub.R]([Ncl.sub.R]([Nint.sub.R](D)))

(iii) N semiopen structure ring (NSemiOSR), if D [??] [Ncl.sub.R]([Nint.sub.R](D))

(iv) N preopen structure ring (NPreOSR), if D [??] [Nint.sub.R]([Ncl.sub.R](D))

(v) N[beta]-open structure ring (N[beta]OSR), if D C [Ncl.sub.R]([Nint.sub.R]([Ncl.sub.R](D)))

Note 3.1. Let (R, S) be a NSR space. Then the complement of a NRegOSR (resp. H[alpha]OSR, NSemiOSR, NPreOSR and N[beta]OSR) is a N regular closed structure ring(NRegCSR) (resp. N[alpha]-closed structure ring(N[alpha]CSR), N semiclosed structure ring(NSemiCSR), H preclosed structure ring(NPreCSR), N[beta]-closed structure ring(N[beta]CSR)).

Definition 3.5. The NSR preinterior and NSR preclosure of NR D of a NSR space are defined and denoted by

(i) [Npint.sub.R](D) = [union]{H : H is a NPreOSR in (R, S) and H [??] D}.

(ii) [Npcl.sub.R](D) = [intersection]{H : H is a NPreCSR in (R, S) and D [??] H}.

Remark 3.1. For any NR D of a NSR space (R, S), then

(i) [Npint.sub.R](D) = D if and only if D is a NPreOSR.

(ii) [Npcl.sub.R](D) = D if and only if D is a NPreCSR.

(iii) [Nint.sub.R](D) [??] [Npint.sub.R](D) [??] D [??] [Npcl.sub.R](D) [??] [Ncl.sub.R](D)

Definition 3.6. A NR D of a NSR space (R, S) is called a H strongly preopen structure ring (Nstrongly PreOSR), if D [??] [Nint.sub.R]([Npcl.sub.R](D)). The complement of a Nstrongly PreOSR is a H strongly preclosed structure ring(briefly Nstrongly PreCSR).

Definition 3.7. The NSR strongly preinterior and NSR strongly preclosure of of NR D of a NSR space are defined and denoted by

(i) [Nspint.sub.R](D) = [union]{H : H is a Nstrongly PreOSR in (R, S) and H [??] D}.

(ii) [Nspcl.sub.R](D) = [intersection]{H : H is a Nstrongly PreCSR in (R, S) and D [??] H}.

Remark 3.2. For any NR D of a NSR space (R, S), then

(i) [Nspint.sub.R](D) = D if and only if D is a Nstrongly PreOSR.

(ii) [Nspcl.sub.R](D) = D if and only if D is a Nstrongly PreCSR.

(iii) [Nint.sub.R](D) [??] [Nspint.sub.R](D) [??] D [??] [Nspcl.sub.R](D) [??] [Ncl.sub.R](D)

Proposition 3.2. A NR of a NSR space (R, S) is a H[alpha]OSR if and only if it is both NSemiOSR and Nstrongly PreOSR.

Definition 3.8. Let ([R.sub.1], [S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. A function [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) is called a NSR

(i) contra continuous function(NSR--CCF) if [[phi].sup.-1](U) is a NOR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]).

(ii) contra [alpha]-continuous function (NSR--C[alpha]CF) if [[phi].sup.-1](U) is a N[alpha]OR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]).

(iii) contra precontinuous function(NSR--CpreCF) if [[phi].sup.-1](U) is a NPreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]).

(iv) contra strongly precontinuous function (NCR--CStrpreCF) if [[phi].sup.-1](U) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]).

Proposition 3.3. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) be a function. If [phi] is a NSR--CCF, then [phi] is a NSR--C[alpha]CF.

Proof:

Let U be a NCR in ([R.sub.2], [S.sub.2]). Since [phi] is NSR--CCF, [[phi].sup.-1](U) is a NOR in ([R.sub.1] [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). By Remark 3.1(iii), [mathematical expression not reproducible].

Since [[phi].sup.-1](U) is a NOR in ([R.sub.1], [S.sub.1]),[mathematical expression not reproducible]. Hence, [[phi].sup.-1](U) is a NSemiOSR in ([R.sub.1], [S.sub.1]). By Remark 3.1 (iii),[mathematical expression not reproducible]. Taking interior on both sides, [mathematical expression not reproducible]. Since [[phi].sup.-1](U) is a NOR in ([R.sub.1], [S.sub.1]),[mathematical expression not reproducible]. Hence, [[phi].sup.-1](U) is a Kstrongly PreOSR in ([R.sub.1], [S.sub.1]). Therefore, [[phi].sup.-1](U) is both NSemiOSR and Nstrongly PreOSR in ([R.sub.1], [S.sub.1]). By Proposition 3.1, [[phi].sup.-1]([phi]) is a N[alpha]OSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Hence, [phi] is a NSR--C[alpha]CF.

Proposition 3.4. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) ([R.sub.2], [S.sub.2]) be a function. If [phi] is a NSR--C[alpha]CF, then [phi] is a NSR--CStrpreCF.

Proof:

Let U be any NCR in ([R.sub.2], [S.sub.2]). Since [phi] is a NCR- C[alpha]CF, [[phi].sup.-1](U) is a N[alpha]OSR in ([R.sub.1] [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). By Proposition 3.1, [[phi].sup.-1](U) is both NSemiOSR and Nstrongly PreOSR in ([R.sub.1] [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Therefore, [[phi].sup.-1](U) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Hence [phi] is a NSR--CStrpreCF.

Proposition 3.5. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) ([R.sub.2], [S.sub.2]) be a function. If [phi] is a NSR--CStrpreCF, then [phi] is a NSR--CpreCF.

Proof:

Let U be any NCR in ([R.sub.2], [S.sub.2]). Since [phi] is a NSR--CStrpreCF, [[phi].sup.-1](U) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]), that is,

[mathematical expression not reproducible] (3.1)

By Remark 3.1(iii),

[mathematical expression not reproducible] (3.2)

Substitute (3.2) in (3.1), we get [mathematical expression not reproducible]. Therefore, [[phi].sup.-1](U) is a NPreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Hence, [phi] is a NSR--CpreCF.

Proposition 3.6. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) be a function. If [phi] is a NSR--CCF, then [phi] is a NSR--CStrpreCF.

Proof:

Let U be any NCR in ([R.sub.2], [S.sub.2]). Since [phi] is a NSR--CCF, [[phi].sup.-1](U) is a NOR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]), that is, [mathematical expression not reproducible]. By Remark 3.1(iii),

[mathematical expression not reproducible] (3.3)

Taking interior on both sides in (3.3),

[mathematical expression not reproducible].

Hence, [[phi].sup.-1] (U) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Thus, [phi] is a NSR--CStrpreCF.

Proposition 3.7. Let ([R.sub.1], [S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) be a function. If [phi] is a NSR--CCF, then [phi] is a NSR--CpreCF.

Proof:

Let U be any NCR in ([R.sub.2], [S.sub.2]). Since [phi] is a NSR--CCF, [[phi].sup.-1](U) is a NOR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]), that is, [mathematical expression not reproducible]. By Remark 3.1(iii),

[mathematical expression not reproducible] (3.4)

Taking interior on both sides in (3.4),

[mathematical expression not reproducible].

Hence, [[phi].sup.-1](U) is a NPreOSR in ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Thus, [phi] is a NSR--CpreCF.

Remark 3.3. The converses of the Proposition 3.2, Proposition 3.3, Proposition 3.4, Proposition 3.5 and Proposition 3.6 need not be true as it is shown in the following example.

Example 3.1. Let R = {a, b, c} be a nonempty set with two binary operations as follows:
+   u   v   w
u   u   v   w
v   v   w   u
w   w   u   v


and
*   u   v   w
u   u   u   u
v   u   v   w
w   u   w   v


Then (R, +, *) is a ring. Define NR's L, M and P as follows:

[mathematical expression not reproducible].

Then [S.sub.1] = {[0.sub.N], [1.sub.N], L, M}, [S.sub.2] = {[0.sub.N], [1.sub.N], P}, [S.sub.3] = {[0.sub.N], [1.sub.N], C(L)} and [S.sub.4] = {[0.sub.N], [1.sub.N], P} are the NSR's on R.

Then the identity function [phi] : (R, [S.sub.2]) [right arrow] (R, [S.sub.3]) is a NSR--CpreCF, but [phi] is neither NSR--CCF nor

NSR--CStrpreCF.

Similarly the identity function [phi] : (R, [S.sub.1]) [right arrow] (R, [S.sub.4]) is a NSR--CStrpreCF but [phi] is neither NSR--CCF nor NSR--C[alpha]CF

Remark 3.4. Clearly the following diagram holds.

Proposition 3.8. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) be a function. Then the following are equivalent.

(i) [phi] is NSR--CStrpreCF.

(ii) [[phi].sup.-1] (U) is a Nstrongly PreCSR in ([R.sub.1], [S.sub.1]), for each NOR U in ([R.sub.2], [S.sub.2]).

Proof:

(i)[??](ii)

Let U be any NOR in ([R.sub.2], [S.sub.2]), then C(U) is a NCR in ([R.sub.2], [S.sub.2]). Since [phi] is a NSR--CStrpreCF, [[phi].sup.-1](C(U)) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR C(U) in ([R.sub.2], [S.sub.2]). By Remark 3.2(i),[mathematical expression not reproducible]. Therefore, [[phi].sup.-1] (U) is a Nstrongly PreCSR in ([R.sub.1], [S.sub.1]), for each NOR U in ([R.sub.2], [S.sub.2]).

(ii)[??](i)

Let C(U) be any NOR in ([R.sub.2], [S.sub.2]). Then U is a NCR in ([R.sub.2], [S.sub.2]). Since [[phi].sup.-1](C(U)) is a Nstrongly PreCSR in ([R.sub.1], [S.sub.1]), for each NOR C(U) in ([R.sub.2], [S.sub.2]). We have, [[phi].sup.-1](U) is a Nstrongly PreOSRin ([R.sub.1], [S.sub.1]), for each NCR U in ([R.sub.2], [S.sub.2]). Hence, [phi] is a NSR--CStrpreCF.

Proposition 3.9. Let ([R.sub.1],[S.sub.1]) and ([R.sub.2], [S.sub.2]) be any two NSR spaces. Let [phi] : ([R.sub.1], [S.sub.1]) [right arrow] ([R.sub.2], [S.sub.2]) be a function. Suppose if one of the following statement hold.

(i) [mathematical expression not reproducible], for each NR V in ([R.sub.2], [S.sub.2]).

(ii) [mathematical expression not reproducible], for each NR V in ([R.sub.2], [S.sub.2]).

(iii) [mathematical expression not reproducible], for each NR V in ([R.sub.1], [S.sub.1]).

(iv) [mathematical expression not reproducible], for each NPreOSR V in ([R.sub.1], [S.sub.1]).

Then, [phi] is a NSR--CStrpreCF.

Proof:

(i)^(ii)

Let U be any NR in ([R.sub.2], [S.sub.2]). Then, [mathematical expression not reproducible] By taking complement on both sides,

[mathematical expression not reproducible]

[mathematical expression not reproducible]

Therefore, [mathematical expression not reproducible], for each NR V = C(U) in ([R.sub.2], [S.sub.2]). (ii)[??](iii)

Let U be any NR in ([R.sub.2], [S.sub.2]). Let V be any NR in ([R.sub.1], [S.sub.1]) such that U = [phi](V). Then V [??] [[phi].sup.-1](U). By (ii), [mathematical expression not reproducible]. We have

[mathematical expression not reproducible]

Therefore, [mathematical expression not reproducible], for each NR V in ([R.sub.1], [S.sub.1]). (iii)[??](iv)

Let V be any NPreOSR in ([R.sub.1], [S.sub.1]). Then [mathematical expression not reproducible]. By (iii),

[mathematical expression not reproducible].

Therefore, [mathematical expression not reproducible], for each NPreOSR V in ([R.sub.1], [S.sub.1]).

Suppose that (iv) holds. Let U be any NOR in ([R.sub.2], [S.sub.2]). Then [mathematical expression not reproducible] is a NPreOSR in ([R.sub.1], [S.sub.1]). By (iv),

[mathematical expression not reproducible].

We have, [mathematical expression not reproducible].

Then [mathematical expression not reproducible]. This implies that [[phi].sup.-1](U) is a NPreCSR in ([R.sub.1], [S.sub.1]). Taking complement on both sides, [mathematical expression not reproducible]. This implies that [mathematical expression not reproducible]. Therefore [[phi].sup.-1](C(U)) is a Nstrongly PreOSR in ([R.sub.1], [S.sub.1]), for each NCR C(U) in ([R.sub.2], [S.sub.2]). Hence, [phi] is a NSR--CStrpreCF.

References

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[2] D.Coker, An Introduction to Intuitionistic Fuzzy Topological Spaces,Fuzzy Sets and Systems, 88(1997), No. 1, 81-89.

[3] D. Coker and M.Demirci, On Intuitionistic Fuzzy Points, Notes IFS 1(1995), no.2, 79-84.

[4] R. Dhavaseelan and S. Jafari, Generalized Neutrosophic closed sets, In New Trends in Neutrosophic Theory and Application, F. Smarandache and S. Pramanik (Editors), Pons Editions, Brussels, Belgium, Vol. 2(2018), 261-274.

[5] E. Ekici and E. Kerre, On fuzzy continuities, Advanced in Fuzzy Mathematics, 1(2006), 35-44.

[6] J. K. Jeon, Y. J. Yun, and J. H. Park, Intutionistic fuzzy [alpha]-continuity and Intuitionistic fuzzy precontinuity, Int. J. Math. Sci., 19(2005), 3091-3101

[7] B. Krteska and E. Ekici, Fuzzy contra strong precontinuity, Indian J. Math. 50(1)(2008), 149-161.

[8] B. Krteska and E. Ekici, Intuitionistic fuzzy contra precontinuity, Filomat 21 (2) (2007), 273-284.

[9] K. Meena and V. Thomas, Intuitionistic L-Fuzzy subrings, International mathematical forem, Vol. 6, 2011, 2561-2572.

[10] R. N. Devi, E. Roja and M. K. Uma, Intuitionistic fuzzy exterior spaces via rings, Annals of Fuzzy Mathematics and Informatics, Volume 9 (2015), No. 1, 141-159.

[11] R. N. Devi, E. Roja and M. K. Uma, Basic Compactness and Extremal Compactness in Intuitionistic Fuzzy Structure Ring Spaces, Annals of Fuzzy Mathematics and Informatics, Volume 23 (2015), No. 3, 643-660.

[12] R. N. Devi and S. E. T. Mary, Contra Strong Precontinuity in Intuitionistic fuzzy Structure ring spaces, The Journal of Fuzzy Mathematics, (2017) (Accepted).

[13] A. A. Salama and S. A. Alblowi, Neutrosophic Set and Neutrosophic Topological Spaces, IOSR Journal of Mathematics, 3 (4), 2012, 31-35.

[14] F. Smarandache, Neutrosophy and Neutrosophic Logic, First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability, and Statistics University of New Mexico, Gallup, NM 87301, USA(2002), smarand@unm.edu

[15] F. Smarandache. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neu-trosophic Probability. American Research Press, Rehoboth, NM, 1999.

[16] L. A. Zadeh, Fuzzy Sets, Infor. and Control, 9 (1965), 338-353.

Received: Juanary 22 2019. Accepted: March 20, 2019

R. Narmada Devi (1), R. Dhavaseelan (2) and S. Jafari (3)

(1) Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Chennai, India.

E-mail: narmadadevi23@gmail.com

(2) Department of Mathematics, Sona College of Technology Salem-636005,Tamil Nadu, India.

E-mail: dhavaseelan.r@gmail.com

(3) Department of Mathematics, College of Vestsjaelland South, Herrestraede 11, 4200 Slagelse, Denmark.

E-mail: jafaripersia@gmail.com

(*) Correspondence: Author (narmadadevi23@gmail.com)
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