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J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011)

A Note On Fuzzy Strongly α -Preirresolute Mappings M. SHUKLA1 and P. SHUKLA2 1

Department of Applied Mathematics, Gayan Ganga Institute of Engineering and Technology Jabalpur – 482011 M.P. India 2 Directorate of Weed Science Research, Maharajpur, Jabalpur – 482004 M.P. India ABSTRACT In this note we have introduced and studied the concept of fuzzy strongly α-preirresolute mappings on fuzzy topological spaces. Some properties and several characterizations of this type of functions are obtained. Keywords and phrases: Fuzzy strongly ߙ-preirresolute, Fuzzy ߙ-continuous, Fuzzy ߚ-open sets, Fuzzy ߙ-open sets. AMS Subject Classification: 54 A.

1. INTRODUCTION The notion of fuzzy set was introduced by L. A. Zadeh in his classical paper10. C. L. Chang3 has extended the concept of topology from a collection of crisp sets and generalized the theory of Topological spaces in fuzzy setting. In this article we have studied properties of fuzzy strongly α-preirresolute maps and established equivalent condition for the map to be fuzzy strongly α-preirresolute. 2. PRELIMINARIES Let be a non-empty crisp set and let be a collection of fuzzy sets on satisfying the following conditions:

(i) 0,1∈ , where 0: , denotes the null fuzzy sets and 1: denotes the whole fuzzy set. (ii) Arbitrary union of members of is a member of . (iii) Finite intersection of members of is a member of . Then is called fuzzy topology on and the pair ( , ) is called fuzzy topological space. Let , be a fuzzy topological space. The members of the collection τ are called fuzzy open sets in the space . A fuzzy set : [0,1] is called a fuzzy closed set in the space .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

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M. Shukla, et al., J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011)

Definition 2.1: Let , be a fuzzy topological space. A fuzzy set in the space is called (i) (ii) (iii) (iv) (v)

Fuzzy semi-open1 if , Fuzzy semi-closed1 if

, Fuzzy preopen if4

, Fuzzy pre-closed if4 , Fuzzy -open if4 ,

Fuzzy -closed if4 , (vii) Fuzzy -open if9

. (viii) Fuzzy -closed if9 . (vi)

Definition 2.2: A function : , , ᇹ is called Fuzzy ß™-continuous if4 ିଵ is fuzzy -open in for every fuzzy open set of . (ii) Fuzzy ßš-continuous if9 ିଵ is fuzzy -open in for every fuzzy open set of . (iii) Fuzzy ß™-irresolute if6 ିଵ is fuzzy -open in for every fuzzy -open set of . (i)

3. FUZZY STRONGLY ࢝PREIRRESOLUTE MAPPINGS Definition 3.1: A mapping : , , ᇹ is called fuzzy strongly ß™preirresolute if ିଵ is fuzzy -open in for every fuzzy -open set of .

Example 3.2: Let = { ଵ , ଶ , = { , } a n d , be fuzzy sets in and be a fuzzy set in respectively defined as ( ଵ ) = 0.3, ( ଶ ) = 0.6, = 0.4 and = 0 . 6 . Let Ď„ = { 0 , ,1} a n d Ď„ '= {0, ,1} be the fuzzy topologies on sets and respectively. The map : defined as , 1,2 is fuzzy strongly preirresolute. Since each fuzzy open set is fuzzy -open, it follows that every fuzzy strongly -preirresolute map is fuzzy -continuous. However converse may not be true. In the following example we see that a fuzzy continuous map may not be fuzzy strongly -preirresolute. Example 3.3: Let = { ଵ , ଶ , = { , } a n d , be fuzzy sets in and respectively defined as ( ଵ ) = 0.5, ( ଶ ) = 0.6, ( ) = 0.7 and ( ) = 0.8. Let Ď„ = { 0 , ,1} a n d Ď„ '= {0, ,1} be the fuzzy topologies on sets and respectively. The map : defined as , 1,2 is fuzzy -continuous. The fuzzy set in defined as ( ) = 0.2, ( ) = 0.3 is fuzzy -open in but ( is not fuzzy -open in . Hence the map : is fuzzy continuous but not fuzzy strongly preirresolute. Theorem 3.4: Let and be fuzzy topological spaces and : be a map. The following statements are equivalent (i) is fuzzy strongly -preirresolute. (ii) For each fuzzy point ்௼ in and each fuzzy -open set in containing ்௼ , there exists a fuzzy -open set in containing ்௼ such that .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

M. Shukla, et al., J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011)

(iii) For each fuzzy -closed set in , ିଵ is fuzzy -closed set in . (iv) For each fuzzy set in ,

. (v) For each fuzzy set in ,

ିଵ ିଵ . (vi) For each fuzzy set in , ିଵ ିଵ . (vii) For each fuzzy set in , . (viii) For each fuzzy set in ,

. Proof: (i)⇒(ii). Let : be a fuzzy strongly -preirresolute. Let ்௼ where and 0 1, be a fuzzy point in and let be a fuzzy -open set in containing the fuzzy point ்௼ . Since, ்௼

, we have ! , i.e. = contains the fuzzy point ்௼ . Moreover, since is fuzzy strongly -preirresolute, so is fuzzy -open set in , containing the fuzzy point ்௼ and . (ii)⇒( i ) . Let be a fuzzy -open set in . For and 0 < ≤ 1, let ்௼ be a fuzzy point in . Then contains ்௼ and so by given condition (ii) there exists a fuzzy -open set in containing the fuzzy point ்௼ and ! . This implies, ! ( and hence ( contains the fuzzy point ்௼ . Thus each fuzzy point of is also a fuzzy point of

. This shows that , i.e.

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is a fuzzy -open set in . Thus : is fuzzy strongly -preirresolute.

(i)⇒(i ii) . Let be a fuzzy -closed set in . Then ௖ = 1- is fuzzy -open set in . Since : is fuzzy strongly -preirresolute, ௖ 1- is fuzzy -open set in . This implies, = 1- ௖ =1 ௖ is fuzzy -closed set in . ( iii) ⇒(i) . Let be a fuzzy -open set in . Then ௖ 1- is fuzzy -closed set in . Therefore by given condition (iii), ௖ = 1- is fuzzy -closed set in . Hence = 1- ௖ = 1- ௖ is fuzzy -open in . Thus : is fuzzy strongly -preirresolute. ( iii) ⇒( i v ) . Let be a fuzzy set in . Since ! we have ! . Now is a fuzzy -closed set in . Hence by given condition (iii), ( is fuzzy -closed set in containing . Since is the smallest fuzzy -closed set containing , it follows that

! ))). This implies ! .

(iv)⇒(iii). Let be a fuzzy -closed set in . Then by given condition (iv), we have

. This implies, ( ( . Since , we deduce that = . As ିଵ is fuzzy -closed set in , it follows that ିଵ is fuzzy -closed set in . (iv)⇒(v). Let be a fuzzy set in . Then by given condition (iv), ( . This

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

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M. Shukla, et al., J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011)

implies . (v)⇒(iv). Let be a fuzzy set in . Then by given condition (v), ( ! ( . This implies, ! ( ))), and hence ! . (i)⇒(vi). Let be a fuzzy set in . Since is a fuzzy -open set in , by given condition (i), is fuzzy preopen set in . Hence we have ( ( ))). Since, ( ( ))) ( , we find that . (vi)⇒(i). Let be a fuzzy -open set in . Then we have . Therefore by given condition (vi), , i.e. ) ! ିଵ . Since , we get that ( = ). Hence ( is fuzzy -open set in . Thus : is a fuzzy strongly -preirresolute map. (iii) ⇒(vii). Let be a fuzzy set in . Then is a fuzzy -closed set in . From given condition (iii), is a fuzzy -closed set in . This implies, " ))))" ))) i.e., ( ! . (vii) ⇒(viii). Let be a fuzzy set in . Then is a fuzzy set in . From given condition (vii) we have, ! ( )) ))! ))). This implies,

! " # ! . Thus we have,

.

(viii)⇒(iii). Let be a fuzzy -closed set in . Then is a fuzzy set in . From given condition (viii) we have, ! ! ( . This implies ! ! . Thus ) is a fuzzy -closed set in .

Theorem 3.5: Let and be fuzzy topological spaces and : be a bijective map. Then is fuzzy strongly preirresolute iff for each fuzzy set in , . Proof: Let : be a bijective map. Suppose is fuzzy strongly -preirresolute. If is a fuzzy set in then is a fuzzy set in . Since is fuzzy strongly preirresolute, from Theorem 3.4 we have, ିଵ ିଵ .

Since is one-one, - = . This shows that . Further since is onto we have, =

. Thus .

Conversely let be a fuzzy -open set in . Then = . Now is a fuzzy set in , from hypothesis,

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

M. Shukla, et al., J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011) ିଵ ିଵ .

Since is onto, = . Therefore " . Further since is one-one we have,

ିଵ ିଵ ିଵ

ିଵ .

As ! , we deduce that . Thus is a fuzzy -open set in . Hence : is fuzzy strongly -preirresolute map. Theorem 3.6: Let , and $ be fuzzy topological spaces and let : and %: $ be maps. If is fuzzy -irresolute and % is fuzzy strongly -preirresolute then %& : $ is fuzzy strongly -preirresolute. Proof: Let ߣ be a fuzzy -open set in $. Since %: $ is fuzzy strongly -preirresolute, ିଵ (ߣ) is fuzzy -open set in . Further, since : is fuzzy -irresolute, ିଵ = %& ିଵ ߣáˆť is fuzzy -open set in . Hence %& ିଵ ߣáˆť is fuzzy -open set in , for each fuzzy -open set ߣ in $. Thus %& : $ is fuzzy strongly -preirresolute.

Theorem 3.7: Let , and $ be fuzzy topological spaces and let : and %: $ be maps. If is fuzzy strongly -preirresolute and % is fuzzy -continuous then %& : $ is fuzzy -continuous. Proof: As above. Corollary 3.8: Let , and be fuzzy topological spaces. Let : and : be projection maps. If :

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is fuzzy strongly -preirresolute map then & : and '& : $ are fuzzy continuous maps. Proof: Since the projection maps : ( $ , ': ( $ $ are fuzzy continuous maps, it follows from corollary 3.8 that the composite maps #$ : and %$ : & are fuzzy -continuous maps. Proposition 3.9: Let and be fuzzy topological spaces such that is product related to . Suppose : is a map and %: → ( is the graph of map . If % is fuzzy strongly -preirresolute then is fuzzy strongly -preirresolute. Proof: Let be a fuzzy -open set in . Since is product related to , we have,

(1( = 1( ! 1 ( in ( . Thus 1( is a fuzzy -open set in ( . Since %: → ( is fuzzy strongly -preirresolute map, it follows that ିଵ ' 1 ( is a fuzzy -open set in . This implies ) → is fuzzy strongly -preirresolute map. Theorem 3.10: Let and , 1,2 be fuzzy topological spaces such that is product related to ଶ and ଵ is product related to ଶ . If ଵ ( ଶ : ( ଶ ଵ ( ଶ is fuzzy strongly -preirresolute than ଵ : ଵ and ଶ : ଶ ଶ are fuzzy strongly -preirresolute. Proof: Let be a fuzzy -open set in ଵ . Since ଵ is product related to ଶ , ( 1 is fuzzy -open set in ଵ ( ଶ . Since ଵ ( ଶ : ( ଶ ଵ ( ଶ is fuzzy strongly preirresolute, we have ଵ ( ଶ ିଵ ( 1 = ( 1 is fuzzy -open set in ( ଶ .

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

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M. Shukla, et al., J. Comp. & Math. Sci. Vol.2 (5), 771-776 (2011)

Further since is product related to ଶ , we have ( 1 = ( 1 ! ( 1. This implies, !

. Thus ( is fuzzy -open set in . Hence ଵ : ଵ is fuzzy strongly -preirresolute map. By a similarly argument we can show that ଶ : ଶ ଶ is fuzzy strongly -preirresolute map. REFERENCES 1. Azad K.K., On fuzzy semi continuity, fuzzy almost continuity and fuzzy weak continuity. J. Math. Anal. Appl. 82, 1432 (1981). 2. Beceren Y. and Nori T., Strongly preirresolute functions and strongly preirresolute functions. J. Pure Math., 18, 1-7 (2001). 3. Beceren Y. and Nori T., -preirresolute functions and -preirresolute functions. Demonstration Math 34, 207-13 (2003).

4. Bin Shahana A.S., On fuzzy strong semi-continuity and fuzzy pre continuity. Fuzzy Sets and Systems. 44, 303-308 (1991). 5. Chang C.L., Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182-190 (1968). 6. Prasad R., Thakur S.S. and Saraf R.K., Fuzzy -irresolute mapping. J. Fuzzy Math., 2, No. 2, 335-339 (1994). 7. Shukla M. and Shukla p. Fuzzy preirresolute Mapping Acta Cinecia Indica (accepted). 8. Shrivastava M. Maitra J.K. and Shukla M., On fuzzy strongly pre continuous mappings, VISLESANA, 10-B (2), 109117 (2006). 9. Thakur S.S. and Singh S., On fuzzy semi-preopen sets and fuzzy semiprecontinuity., Fuzzy Sets and Systems., 98, 383-391 (1998). 10. Yalvac, T. H., Semi interior and semiinterior of fuzzy sets. J. Math. Anal. Appl., 132, 365-364 (1988). 11. Zadeh L. A., Fuzzy Sets, Inform. and Control, 8, 338-353 (1965).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 5, 31 October, 2011 Pages (693-779)

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