Fuzzy Pairwise Strongly Pre-Continuous Mappings

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Vol. 3, Issue 6, June 2014

Copyright to IJIRSET www.ijirset.com 13927

Fuzzy Pairwise Strongly Pre-Continuous

Mappings

Madhulika Shukla

Associate professor, Department of Applied Mathematics, GGITS Engineering College, Jabalpur (M.P.) India

ABSTARCT: We define and characterize a fuzzy pairwise strongly pre-continuous mappings on a fuzzy bitopological space. We investigate some of their properties. We establish some equivalent conditions of fuzzy pairwise strongly pre-continuous mappings on a fuzzy bitopological space.

KEYWORDS: ( , )-fuzzy preopen, ( , )-fuzzy preclosed, ( , )-fuzzy semi-open, ( , )-fuzzy semi-closed, ( , ) -fuzzy pairwise precontinuous, ( , )-fuzzy pairwise semicontinuous, ( , )-fuzzy pairwise stongly pre-continuous, ( , )-fuzzy pairwise strongly preclosed.

I.

INTRODUCTION

In 1981, K.K. Azad [4] introduced the concept of semi-open sets in fuzzy topology. A.S. Bin Shahana [1] has defined the concept of fuzzy pre-open sets in fuzzy topological spaces.

In 1989, A. Kandil [5] introduced the notation of fuzzy bitopological space. Further in 1996, S.S. Thakur and R. Malviya [9] defined fuzzy semi-open and fuzzy semi-continuous in fuzzy bitopological space. Sampath kumar [10] defined a ( , )-fuzzy pre-open set and characterized a fuzzy pairwise precontinuous mappings on a fuzzy bitopological space. Further M. Shrivastava, J.K.m maitra And M. Shukla [7] in 2006 defined fuzzy strongly pre-continuous mapping in fuzzy topological space.

In this article we have established equivalent conditions for a mapping to be fuzzy pairwise strongly pre-continuous mapping in fuzzy bitopological space. Further we have studied some properties of fuzzy pairwise pre-continuous mapping.

II.

PRELIMINARIES

Let be a set and let and be fuzzy topologies on . Then we call ( , , ) a fuzzy bitopological space [fbts].

A mapping : ( , , )→( , ∗_, ∗_{) is fuzzy pairwise continuous [} _{] if the induced 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.

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(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 ≤ − ( − ( ))[9]. (ii) a ( , )-fuzzy semi-closed [( , )− ] set on if − − ( ) ≤ [9]. (iii) a ( , )-fuzzy pre-open [( , )− ] set on if ≤ − ( − ( ))[10]. (iv) a ( , )-fuzzy pre-closed [( , )− ] set on if − ( − ( ))≤ [10]. Definition 2.2. Let be a fuzzy set on a .

(i) The ( , )-semi interior of , [( , )− ] is ∪{ : ≤ , is a ( , )− set} [9]. (ii) The ( , )-semi closure of , [( , )− ] is

∩{ : ≥ , is a ( , )− set} [9]. (iii) The ( , )-pre interior of , [( , )− ( )] is

∪{ : ≤ , is a ( , )− set} [10]. (iv) The ( , )-pre clouser of , [( , )− ( )] is

∩{ : ≥ , is a ( , )− set} [10].

Definition 2.3. Let : ( , , )→( , ∗_, ∗_{) be a mapping. Then is called;}

(i) fuzzy pairwise semi-continuous [ continuous ] mapping if ( ) is a ( , )− set on for each

∗₋

set on Y [9].

(ii) fuzzy pairwise pre-continuous [ continuous] mapping if ( ) is a ( , )− set on for each ∗−

set on Y [10].

(iii)fuzzy pairwise -continuous [ open] mapping if ( ) is a ( , )− set on for each ∗− set on Y [9].

It is clear that every mapping is a and mappings on . But the converse may true

in general.

III. FUZZY PAIRWISE STRONGLY PRE-CONTINUOUS MAPPINGS

Definition 3.1. Let : ( , , )→( , ∗_, ∗_{) be a mapping. Then is called a fuzzy pairwise strongly pre-continuous}

[ continuous] mapping if ( ) is a ( , )− set on for each ( ∗, ∗₎₋

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Since any fuzzy open set is a fuzzy semi-open, if follows that every fuzzy pairwise strongly precontinuous map ( continuous) is fuzzy pairwise precontinuous ( continuous). However, converse may not be true in general. We have the following example.

Example 3.2. Let = { , } and = { , } and , , ∗_, ∗_{be fuzzy sets defined as follows.}

( ) = 0.5, ( ) = 0.6,

( ) = 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 precontinuous [ -continuous] but not fuzzy pairwise strongly precontinuous [ -continuous] mapping.

Theorem 3.3. Let : ( , , )→( , ∗_, ∗_{) be a mapping. Then the following statement are equivalent;}

(i) is fuzzy pairwise strongly pre-continuous [ continuous] mapping. (ii) The inverse image of each ( ∗, ∗_{) –} _{set on is a}₍ _{, )}_- _{set on} _.

(iii) ( , − ( ))≤ ∗_, ∗ ₋_{scl( (}_μ₎₎_{for each fuzzy set on} _.

(iv) ( , )− ( ) ≤ ∗_, ∗ ₋_scl(_ϑ₎ _{for each fuzzy set on} _.

(v) ∗, ∗ ₋_sint(_ϑ₎ _≤ _, ₋ ₍ _{( ))}_{for each fuzzy set on} _.

Proof. (i) ⇒ (ii): Let be a ( ∗, ∗_)- _{set on} _._Then _{is a (} ∗_, ∗_)- _{set on} _._{Since, is} _continuous,

( ) = ( ( )) is a ( , ) – set on . Hence, ( ) is a ( , )- set on .

(ii) ⇒ (iii): Let is a fuzzy set on . Then ∗_, ∗ ₋_{scl (}_μ₎ _{is a (} _{, )}_- _{set on} _._{Thus (} _{, )}₋ _{( )}_≤

, − ( ) ≤ , − ( ( ∗_, ∗ ₋_scl ₍_μ_{) ))}₌ ₍ ∗_, ∗ ₋_scl _{( ) ).}_Hence, ₍ _, ₋

( ))≤ ( ∗_, ∗ ₋_pcl _{( ) )} _≤ ∗_, ∗ ₋_scl₍ _{( )).}

(iii) ⇒ (iv): Let be a fuzzy set on . Then ( , -p ( ( )))≤ ∗, ∗ - ( ( ( )))≤ ∗, ∗ − ( ). Hence

, − ( ( ))≤ ( , − ( ) )≤ ( ∗, ∗ - ( )).

(iv) ⇒(v). Let be a fuzzy set on . Then , - ( ( ))≤ ( ∗_, ∗ _- ₍ _))._Hence, ₍ ∗_, ∗

-( )) = (( ∗_, ∗ ₋ ₍ ₎₎ ₎_≤₍ _, ₋ ₍ ₍ ₎₎₎₌ _, _- _{( ) .}

(v) ⇒( ). Let be a ∗, ∗

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Hence ( ) is a , - set on and therefore, is continuous function.

Theorem 3.4. Let : ( , , )→( , ∗_, ∗₎

be a bijection. is continuous mapping if and only if for each fuzzy set on .

∗, ∗ ₋_sint ₍_μ₎ _≤ _, ₋ _{( ) .}

Proof: Let be a fuzzy set on . Then, by Theorem 3.3,

( ∗_, ∗ ₋_{sint( (}_μ₎₎₎_≤ _, ₋ ₍_μ_{) .}

Since is a bijection,

∗_, ∗ ₋_sint ₍_μ_{) =} ∗_, ∗ ₋_pint ₍_μ₎ _≤ ₍ _, ₋ _{( ))}_.

Conversely, let be a fuzzy set on . Then

∗_, ∗ ₋_sint _{( )} _≤ _, ₋ _{( ) .}

Recall that is a bijection. Hence

∗_, ∗ ₋_sint(_ϑ_{) =} ∗_, ∗ ₋_sint( ₍_ϑ_{) )}_≤

( , − ( (ϑ))).

and ( ∗_, ∗ ₋_sint(_ϑ₎₎_≤ ₍ _, ₋ ₍_ϑ_{) )}

= , − (ϑ) .

Therefore, by theorem 3.3, is continuous mapping.

Definition 3.5. Let : ( , , )→( , ∗_, ∗₎_{be a mapping. Then is called}

(i) a fuzzy Pairwise strongly pre-open [ open] mapping if ( ) is a ∗, ∗ ₋ _{set on for each}

, − set on .

(ii) a fuzzy Pairwise strongly pre-closed [ closed] mapping if ( ) is a ∗, ∗ ₋ _{set on for each}

, - set on .

Theorem 3.6. Let : ( , , )→( , ∗_, ∗₎_{be a mapping. Then the following statement are equivalent:}

(i) is open mapping.

(ii) ( , − ( ))≤ ∗, ∗ −sint( ( )) for each fuzzy set on .

(iii) , − ( ( ))≤ ( ∗_, ∗ ₋_{sint (}_ϑ₎₎_{for each fuzzy set}_ϑ_on _.

Proof. (i)⇒( ). Let be a fuzzy set on . Then ( , − ( )) is a ∗, ∗ ₋

set on and , −

( ) ≤ ( ). Hence

, − ( ) = ∗_, ∗ ₋_sint( _, ₋ _{( )}

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Copyright to IJIRSET www.ijirset.com 13931 ≤ ∗_, ∗ ₋ _{( ) .}

( )⇒(iii). Let ϑ be a fuzzy set on . Then

, − ( ) ≤ ∗_, ∗ ₋_{sint( (} ₍_ϑ₎₎₎

≤ ∗_, ∗ ₋_{sint (}_ϑ_).

Hence , − (ϑ) ≤ , − (ϑ)

≤ ∗_, ∗ ₋_sint_ϑ _.

(iii)⇒( ). Let be a , − set on . Then

= , − ( )≤ , − ( ( ) )

≤ ∗_, ∗ ₋_sint ₍_μ_{) .}

We have ( )≤ ( ( ∗_, ∗ ₋_sint(

( ))))≤ ∗_, ∗ ₋_sint(

( )).

Hence ( ) = ∗, ∗ ₋_sint _{( ) .}

Consequently, ( ) 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 ( )≤ , − ( ) .

Hence

∗, ∗ −scl ( ) ≤ ∗, ∗ −scl , − ( )

= , − ( ) .

Conversely, let be a , - set on . Then

∗, ∗ ₋_{scl( ( ))}_≤ ₍ _, ₋ _{( ))}

= ( ).

Consequently, ( ) is a ∗, ∗ ₋

on and therefore is a -closed mapping.

Theorem 3.8. Let : ( , , )→( , ∗_, ∗₎_{be a bijection. Then the following statements are equivalent:}

(i) is -closed mapping.

(ii) ∗, ∗ ₋_{scl (}_ϑ₎ _≤ _, ₋ ₍ ₍_ϑ₎₎_{for each fuzzy set}_ϑ_on _.

(iii) is -open mapping.

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Proof. (i)⇒(ii). Let ϑ be a fuzzy set on . Then by Theorem 3.7,

∗_, ∗ ₋_{scl( (} ₍_ϑ₎₎₎_≤ ₍ _, ₋ ₍ ₍_ϑ_))).

Hence ( ∗_, ∗ ₋_scl( ₍ ₍_ϑ₎₎₎_≤ _{( (} _, ₋ ₍ ₍_ϑ_)))).

∗_, ∗ ₋_{scl (}_ϑ₎ _≤ _, ₋ ₍_ϑ_{) .}

(ii)⇒(i). Let be a fuzzy set on . Then

∗, ∗ ₋_{scl (}_μ₎ _≤ _, ₋ ₍_μ_{) .}

Hence ∗, ∗ ₋_scl ₍_μ₎ _≤ _, ₋ ₍_μ₎ _.

∗, ∗ ₋_scl ₍_μ₎ _≤ _, ₋ ₍_μ_{) .}

Therefore by the theorem 3.7, is -closed mapping.

(ii)⇒(iii). Let be a fuzzy set on . Then

∗, ∗ ₋_scl( ₎ _≤ _, ₋ ₍ _{) .}

, − ( ) = ( , − ( ( )))

≤ (( ∗_, ∗ ₋_scl( _{)) )}

= ∗, ∗ ₋_{sint ( ) .}

Hence is -open mapping from Theorem 3.6 .

(iii) ⇒(iv). Let be a fuzzy set on . Then

, − ( ) ≤ ∗, ∗ −sint (ϑ) .

Since is a bijection. by Theorem 3.4 , is continuous mapping.

(iv) ⇒(ii). It is clear from Theorem 3.3.

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Copyright to IJIRSET www.ijirset.com 13933 Corollary 3.9. Let : ( , , )→( , ∗_, ∗₎_{be a mapping. Then is} _{closed and} _{continuous if and only if}

, − ( ) = ∗_, ∗ ₋_{scl( (}_μ₎₎

for each fuzzy set on .

Corollary 3.10. Let : ( , , )→( , ∗_, ∗₎_{be a mapping. Then is} _{open and} _{continous if and only if}

( ∗_, ∗ ₋_scl(_ϑ₎₎

= , − ( ( )) for each fuzzy set on .

Theorem 3.11. Let ( , , ), ( , , ), ( , , ) be fuzzy topological spaces. If : → is fuzzy pairwise strongly precontinuous [ continuous] mapping and : → is fuzzy pairwise semicontinuous [ continuous] mapping then : → is fuzzy pairwise precontinuous [ continuous] mapping.

Concluding remark:

1. We have introduced and studied new kind of map fuzzy pairwise Strongly pr-econtinuos maps on fuzzy bitopological spaces.

2. We defined the relation between fuzzy pairwise pre continuous and fuzzy pairwise strongly continuous map. We investigated some of their properties.

3. We proved that the fuzzy pairwise storgly pre-continuous map is stronger form of fuzzy paqairwise pre-continuous map by use of example.

4. We have established some significant properties of fuzzy pairwise strogly pre-continuous maps.

5. We introduce and study new kind of fuzzy pairwise strongly pre-closed and investigate of their properties.

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