J. Comp. & Math. Sci. Vol.3 (6), 631-636 (2012)
Fuzzy Pairwise Strongly ࢝-Continuous Mappings M. SHUKLA Department of Applied Mathematics, Gayan Ganga Insitude of Technology and Sciences, Jabalpur, M. P., INDIA. (Received on: November 22, 2012) ABSTRACT We define and characterize a fuzzy pairwise strongly continuous mappings on a fuzzy bitopological space. We investigate some of their properties. We establish some equivalent condition of fuzzy pairwise strongly -continuous mappings on a fuzzy bitopological space. Keywords: ( ŕŻœ , ŕŻ? -fuzzy open, ( ŕŻœ , ŕŻ? -fuzzy closed, ( ŕŻœ , ŕŻ? fuzzy semi-open, ( ŕŻœ , ŕŻ? -fuzzy semi-closed, ( ŕŻœ , ŕŻ? -fuzzy pairwise -continuous, ( ŕŻœ , ŕŻ? -fuzzy pairwise semicontinuous,( ŕŻœ , ŕŻ? -fuzzy pairwise -open, ( ŕŻœ , ŕŻ? -fuzzy pairwise -closed. AMS Subject Classification: 54 A40.
1. INTRODUCTION A triplet ( , , , where is a non-empty set and and are fuzzy topologies on is called a fuzzy bitopological space and A. Kandil4 initiated the study of such spaces. Further in 1996, S. S. Thakur and R. Malviya7 defined fuzzy semi-open and fuzzy semi-continuous in fuzzy bitopological space. Sampath Kumar5 defined a ( , -fuzzy -open set and characterized a fuzzy pairwise -continuous mappings on a fuzzy bitopological space. In this article we have established equivalent
conditions for a mapping to be fuzzy pairwise strongly -continuous mapping in fuzzy bitopological space. Further we have studied some properties of fuzzy pairwise continuous mapping. 2. PRELIMINARIES Let be a set and let and be fuzzy topologies on . Then we call , , a fuzzy bitopological space [ ]. A mapping : , , , , ) is fuzzy pairwise continuous [ if the induced
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M. Shukla, J. Comp. & Math. Sci. Vol.3 (6), 631-636 (2012)
mapping : , , is fuzzy continuous for 1,2. A mapping : , , , , ) is fuzzy pairwise open [ open] ( fuzzy pairwise closed [ closed]) if the induced mapping
: , , is fuzzy open (fuzzy closed) for 1,2. Notations. (1) Throughout this paper, we take an ordered pair , with , 1,2 and . (2) For simplicity, we abbreviate a -fuzzy open set and a -fuzzy closed set with a - set and a - set respectively. Also, we denote the interior and the closure of for a fuzzy topology – and " # respectively. Definition 2.1. Let be a fuzzy set on a
. Then we call ; (i) a ( , -fuzzy semi-open [( , "
set on if % " # " 7. (ii) a ( , -fuzzy semi-closed [( , "
set on if " & " # ' % 7. (iii) a ( , -fuzzy (-open [( , " ( set on if % " " # " 5. (iv) a ( , -fuzzy (-closed [( , " ( set on if " # " " # % 5. Definition 2.2.5 Let be a fuzzy set on a
.
(1) The ( , -( interior of , [( , " ( is ) *: * % , * is a ( , " ( set}.
(2) The ( , -( closure of , [( , " ( # is + *: * , , * is a ( , " ( set}. Definition 2.3: Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť) כ‏ be a mappings. Then is called; (1) fuzzy pairwise semi-continuous ( mapping if ିଵ is a ( , set on for each ŕŻœâ€Ť כ‏set on Y7. (2) fuzzy pairwise pre-continuous ( ) mapping if ିଵ is a ( , set on for each ŕŻœâ€Ť כ‏set on Y5. (3) fuzzy pairwise -continuous ( ) mapping if ିଵ is a ( , set on for each ŕŻœâ€Ť כ‏set on Y5. It is clear that every mapping is a and mappings on . But the converse may true in general. 3. FUZZY PAIRWISE STRONGLY -CONTINUOUS MAPPINGS In this section we introduce a fuzzy pairwise strongly (-continuous mapping which is stronger form of fuzzy pairwise (continuous mapping. And we characterize a fuzzy strongly (-continuous mapping. Definition 3.1. Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť) כ‏ be a mapping. Then is called a fuzzy pairwise strongly -continuous [ ] mapping if ିଵ is a ( , set on
for each ( ŕŻœâ€Ť כ‏, ŕŻ?‍ כ‏set on Y. Since any fuzzy open set is fuzzy semi-open, if follows that every fuzzy pairwise strongly (-continuous map - ( ] is fuzzy pairwise (-continuous - ( . However converse may not true in general. We have the following example.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 6, 31 December, 2012 Pages (557-663)
M. Shukla, J. Comp. & Math. Sci. Vol.3 (6), 631-636 (2012)
Example 3.2. Let = {. , . and / , / and , , *, , * be fuzzy sets defined as follows. . = 0.2, . 0.3, . = 0.5, . 0.5, * . = 0.2, * . 0.4, / 0.3, / 0.4, and * / = 0.4, * / 0.5. Let 0, , , 1 , 0, *, 1 and 0, , 1 , 0, * , 1 . Then the mapping : , , , , ) defined by (. / , . / is fuzzy pairwise (-continuous [ ( ] but not fuzzy pairwise strongly (-continuous [ ( ] mapping. Theorem 3.3. Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť) כ‏ be a mappings. Then the following statements are equivalent; (i) is fuzzy pairwise strongly (-continuous ( ( mapping. (ii) the inverse image of each ( , ) – set on is a ( , - ( set on . (iii) & , ' " ( # % & , ' " scl Âľ for each fuzzy set on . (iv) ( , " ( #& * ' %
7& , ' " scl 8 9 for each fuzzy set * on .
(v) 7& , ' " sint 8 9 % & , ' " ( * for each fuzzy set * on .
Proof. (i) = (ii): Let * be a ( , )- set on . Then * is a ( , )- set on . Since is ( , * * is a ( , – ( set on . Hence * is a( , - ( set on . (ii) = (iii): Let is a fuzzy set on . Then
7& , ' " scl&f Âľ '9 is a ( , - (
633
set on . Thus ( , " ( # % & , ' " ( # 7 & '9 %
& , ' " ( # & , ' " scl& Âľ ' = & , ' "scl& '. Hence
& , ' " ( # % 7& , ' " scl& '9 % & , ' " scl( .
(iii) = (iv): Let * be a fuzzy set on . Then
& , '-( # * % & , ' # * ))% & , ' " # * . Hence & , ' " ( # * % 7& , ' "
( #& * '9 % (& , '- # * . (iv) = v). Let * be a fuzzy set on . Then & , '-( # * % & , ' # * . Hence, & , '- *
& , ' " # * )% & , ' " ( # * = & , '-( & * '. (vi) = . Let * be a & , '- set on . Then * = 7& , ' "
sint 8 9 % & , ' " ( & * '.
Hence * is a & , '- set on and therefore, is ( function. Theorem 3.4. Let : , , , , be a bijection. is ( mapping if and only if for each fuzzy set on . ŕŻœâ€Ť כ‏, ŕŻ?‍ כ‏sint Âľ ŕŻœ , ŕŻ? .
Proof: Let be a fuzzy set on . Then, by Theorem 3.3, ିଵ ŕŻœâ€Ť כ‏, ŕŻ?‍ כ‏sint Âľ ŕŻœ , ŕŻ? ିଵ Âľ !.
Since is a bijection.
ŕŻœâ€Ť כ‏, ŕŻ?‍ כ‏sint Âľ ିଵ ŕŻœâ€Ť כ‏, ŕŻ?‍ כ‏ Îąint f Âľ f ŕŻœ , ŕŻ? .
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M. Shukla, J. Comp. & Math. Sci. Vol.3 (6), 631-636 (2012)
Conversely, let * be a fuzzy set on . Then & , ' " sint 7 & '9 % 7& , '
" ( & * '9. Recall that is a bijection. Hence & , ' " sint 8 & , ' " sint f& 8 ' % (& , ' " ( 8 . and
& , ' " sint8 % 7& , ' " ( & 8 '9
= & , ' " ( & 8 '. Therefore, by theorem 3.3, is ( function. Definition 3.5. Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť כ‏ be a mapping. Then is called (i) a fuzzy Pairwise strongly (-open [ ( open] mapping if is a & , ' "
set on for each & , ' ( set on . (ii) a fuzzy Pairwise strongly (-closed [ ( closed] mapping if is a & , ' " set on for each & , ' ( set on . Theorem 3.6. Let : , , , , be a mapping. Then the following statements are equivalent: (i) is ( open mapping. (ii) & , ' " ( % & , ' " sint( for each fuzzy set on . (iii) & , ' " ( * %
& , ' " sint 8 for each fuzzy set 8 on .
Proof. (i) = ii . Let be a fuzzy set on . Then & , ' " ( is a & , ' "
set on and 7& , ' " ( 9 %
. Hence 7& , ' " ( 9 & , ' " sint 7& , ' " ( 9) % & , ' "
& '. ii =(iii). Let 8 be a fuzzy set on . Then
7& , ' " ( & * '9 % & , ' "
sint & 8 ' % & , ' " sint 8. Hence & , ' " ( & 8 ' %
? 7& , ' " ( & 8 '9@
% 7& , ' " sint 89.
(iii)= . Let be a & , ' " ( set on . Then & , ' " ( % & , ' " ( & ' . We have % ( & , ' " sint ( % & , ' " sint ( . Hence & , ' " sint& '. Conversely, is a & , ' " set on and therefore, is ( open mapping.
Theorem 3.7. A mapping : , , , , is closed mapping if and only if & , ' " scl % & , ' " ( # for each fuzzy set on . Proof. Let be a fuzzy set on . Then
& , ' " ( # is a & , '- set on
and % 7& , ' " ( # 9. Hence & , ' " scl& ' % & , ' "scl ? 7& , ' " ( # 9@
= 7& , ' " ( # 9.
Journal of Computer and Mathematical Sciences Vol. 3, Issue 6, 31 December, 2012 Pages (557-663)
M. Shukla, J. Comp. & Math. Sci. Vol.3 (6), 631-636 (2012)
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Conversely, let be a & , '- ( set on . Then & , ' " scl f % & , ' " ( # = . Consequently, is a & , ' " ( on and therefore is a ( open mapping.
(ii) =(iii). Let * be a fuzzy set on . Then
7& , ' " scl * 9 % & , '
Theorem 3.8. Let : , , , , be a bijection. Then the following statements are equivalent: (i) is ( closed. (ii) 7& , ' " scl 89 % & , ' " ( # 8 for each fuzzy set 8 on . (iii) is ( mapping. (iv) is ( mapping.
= 7& , ' " sint *9. Hence is ( open mapping from Theorem 3.6. (iii) =(iv). Let * be a fuzzy set on . Then & , ' " ( & * ' %
Proof. (i)=(ii). Let 8 be a fuzzy set on . Then by Theorem 3.6, & , ' " scl 8 % & , ' " ( #& 8 '. Hence
& , ' " scl ( 8 %
& , ' " ( # 8 . Since is a bijection,
7& , ' " scl 89 % & , ' "
( #& 8 '. (ii) =(i). Let be a fuzzy set on . Then
7& , ' " scl& Âľ '9 % & , '
"( #& * '.
By Lemma, & , ' " ( & * ' & , ' " ( # *
% & , ' " scl * #
7& , ' " sint 89. Since is a bijection. by Theorem 3.4, is ( mapping. (iv) =(ii). It is clear from Theorem 3.3.
Corollary 3.9. Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť כ‏ be a mapping. Then is ( closed and
( if and only if 7& , ' " ( # 9 & , ' " scl f Âľ for each fuzzy set on .
"( # 7 & Âľ '9.
Corollary 3.10.Let : , ଵ , ଶ , ŕŹľâ€Ť כ‏, ŕŹśâ€Ť כ‏ be a mapping. Then is ( open and
( if and only if & , ' " scl8 = & , ' " ( # for each fuzzy set * on .
Hence
REFERENCES
? 7& , ' " scl& Âľ '9@ %
?& , ' " ( # 7 & Âľ '9@.
Since is a bijection, & , ' " scl& Âľ ' % 7& , ' " ( #Âľ9. Therefore by the theorem 3.7, is ( closed.
1. Azad K.K., On fuzzy semi continuity, fuzzy almost continuity and fuzzy weak continuity. J. Math. Anal. Appl. 82,1432 (1981). 2. Chang C.L., Fuzzy topological spaces. J. Math. Anal. Appl. 24, 182-190 (1968).
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3. H.S. Lee, J.S. Lee and Y.B. Im, Fuzzy pairwise strong pre-irresolute continuous mappings. J. Appl. Math & Informatics. 28, 1561-1571 (2010). 4. Kandil, Biproximities and fuzzy bitopological spaces, Simon Stevin. 63, 45-66 (1989). 5. Sampath kumar S., On fuzzy pairwise (continuous and fuzzy pairwise precontinuity. Fuzzy Sets and Systems. 62, 231-238 (1994).
6. Shukla M., Fuzzy strongly (-continuous mapping in International Journal of Fuzzy Mathematics & System Vol. 2, No.2, pp 163-169 (2012). 7. Thakur S.S. and Malviya R., Semi-open sets and semi-continuity in fuzzy bitopological spaces. Fuzzy Sets and Systems. 79, 251-256 (1996). 8. Zadeh L.A., Fuzzy Sets, Inform. and Control, 8, 338-353 (1965).
Journal of Computer and Mathematical Sciences Vol. 3, Issue 6, 31 December, 2012 Pages (557-663)