Download INTUTIONISTIC FUZZY PRE SEMI CLOSED SETS Author: D.Amsaveni , M.K.Uma and E.Roja

Int. J. of Mathematical Sciences and Applications,
Vol. 1, No. 3, September 2011
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INTUTIONISTIC FUZZY PRE SEMI CLOSED SETS
D.Amsaveni
[email protected]
M.K.Uma and E.Roja
Sri Sarada College for Women, Salem-636016
Tamilnadu
Abstract
A new class of sets namely, IF pre semi closed sets is introduced in IF topological spaces. This class is
properly placed in between the class of IF semi pre closed sets and the class of IF generalized semi pre
closed sets. Applying IF pre semi closed sets, we introduce and study two new spaces, namely IF pre
semi T 1 spaces, IF semi pre T 1 spaces. We characterize IF pre semi T 1 spaces. We proved that the
2
3
dual of the class of IF pre semi
2
T1 spaces to the class of IF semi pre T1 spaces is the class of IF
2
semi pre
2
T1 spaces. We also introduce and study IF pre semi continuous functions and IF pre semi
3
irresolute functions.
Keywords: IF pre semi closed sets, IF pre semi
T1 spaces, IF semi pre T1 spaces, IF pre semi continuous functions and
2
3
IF pre semi irresolute functions.
1. Introduction
After the introduction of the concept of fuzzy sets by Zadeh [17], several researches were conducted on the generalizations
of the notion of fuzzy set. The concept of “Intuitionistic fuzzy sets” was first published by Atanassov [1] and many works by
the same author and his colleagues appeared in the literature [2-4]. Later this concept was generalized to “Intuitionistic Lfuzzy sets” by Atanassov and stoeva [5]. An introduction to intuitionistic fuzzy topological space was introduced by Dogan
Coker [7]. Several types of fuzzy connectedness in intuitionistic fuzzy topological spaces defined by Coker (1997). The
construction is based on the idea of intuitionistic fuzzy set developed by Atanassov (1983,1986; Atanassov and Stoeva,
1983). Pre semi closed set was introduced by [15]. In this paper a new class of sets namely, IF pre semi closed sets is
introduced for IF topological spaces. This class is properly placed in between the class of IF semi pre closed sets and the
class of IF generalized semi pre closed sets. An applications of IF pre semi closed sets, we introduce and study new spaces
namely, IF pre semi T 1 spaces, IF semi pre T 1 spaces, We proved that the class of IF pre semi T 1 spaces properly
2
3
contains the class of IF semi pre
2
T1 spaces, the class of IF semi pre T1 spaces properly contains the class of IF semi
2
pre
3
T1 spaces. We proved that the dual of the class of IF pre semi T1 spaces to the class of IF semi pre T1 spaces is
2
2
the class of IF semi pre
2
T1 spaces. We also introduce and study IF pre semi continuous functions and IF pre semi irresolute
3
functions.
2. Preliminaries
Definition 2.1.[3] Let
form
A
X be a non empty fixed set. An intuitionistic fuzzy set (IFS for short) A is an object having the
x,  A ( x),  A ( x) : x  X  where the functions  A : X  I and  A : X  I denote the degree of
membership (namely
respectively, and
 A  x  ) and the degree of non membership (namely  A  x  ) of each element x  X
0   A  x    A  x   1 for each x  X .
A  x,  A ,  A .
Definition 2.2.[3] Let X be a non empty set and the IFSs A and B be in the form
A   x,  A ( x),  A ( x ) : x  X  , B   x,  B ( x),  B ( x) : x  X  .Then
Remark 2.1. For the sake of simplicity, we shall use the symbol
(a)
A  B iff  A  x    B  x  and  A  x    B  x  for all x  X ;
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to the set
A,
D.Amsaveni, M.K.Uma and E.Roja
A  B and B  A ;
x,  A ( x),  A ( x) : x  X  ;
(b) A  B iff
(c)
(d)
(e)
(f)
(g)
A 
A  B   x,  A ( x)   B ( x),  A ( x)   B ( x) : x  X  ;
A  B   x,  A ( x)   B ( x),  A ( x)   B ( x) : x  X  ;
[ ] A   x,  A ( x),1   A ( x) : x  X  ;
  A   x,1   A ( x),  A ( x) : x  X  .
Definition 2.3.[7] Let X be a non empty set and let
 Ai 
 x,  
Ai

 Ai :i  J  be an arbitrary family of
( x),   Ai ( x) : x  X ; (b)  Ai 
Definition 2.4.[7] Let X be a non empty fixed set. Then,
Definition 2.5[7] Let
(a) If
is the IFS in
(b) If
B 
 x, 
Ai

IFSs in

X . Then (a)
( x),  Ai ( x) : x  X .
0~   x, 0,1 : x  X  and 1~   x,1, 0 : x  X  .
X and Y be two non empty fixed sets and f : X  Y be a function. Then
y,  B ( y ),  B ( y ) : y  Y  is an IFS in Y, then the pre image of B under f , denoted by f 1  B  ,
1
X defined by f ( B) 

1

1
x, f ( B )( x), f (  B )( x) : x  X .
A   x,  A ( x),  A ( x) : x  X  is an IFS in X , then the image of A under f , denoted by f ( A) , is
the IFS in Y defined by
f ( A)   y, f ( A )( y ),(1  f (1   A ))( y ) : y  Y  where,
 sup  A ( x) if f 1 ( y )  0
f ( A )( y )   x f 1 ( y )
 0,
otherwise,
 inf  A ( x) if f 1 ( y )  0
(1  f (1  A ))( y )   x f 1 ( y )
 1,
otherwise.
for the IFS
A   x,  A ( x),  A ( x ) : x  X  . 
Definition 2.6[7] Let X be a non empty set. An intuitionistic fuzzy topology (IFT for short) on a non empty set X is a
family  of intuitionistic fuzzy sets (IFSs for short) in X satisfying the following axioms: (T1) 0 ~ ,1~  , (T2)
G1  G2  for any G1 , G2  ,
(T3)  Gi   for any arbitrary family Gi : i  J  .

In this case the pair  X ,   is called an intuitionistic fuzzy topological space (IFTS for short) and any IFS in  is
known as an intuitionistic fuzzy open set (IFOS for short) in X .
Definition 2.7.[7] Let X be a non empty set. The complement A of an IFOS A in an IFTS  X ,   is called an
intuitionistic fuzzy closed set (IFCS for short) in X .
Definition 2.8.[7] Let  X ,   be an IFTS and A  x,  A ,  A be an IFS in X . Then the fuzzy interior and fuzzy
closure of A are defined by
cl ( A)    K : K is an IFCS in X and A  K ,
int( A)  G : G is an IFOS in X and G  A.
Remark 2.2.[7] Let  X ,  be an IFTS. cl ( A) is an IFCS and int( A) is an IFOS in X , and
(a) A is an IFCS in X iff cl ( A)  A ; (b) A is an IFOS in X iff int( A)  A .
Proposition 2.1.[7] Let
 X , 
be an
IFTS.
For any IFS A
int( A )  cl ( A).
1476
in
 X ,  ,
we have(a)
cl ( A )  int( A),
(b)
INTUTIONISTIC FUZZY PRE SEMI CLOSED SETS
 X , 
(Y , ) be two IFTSs and let f : X  Y be a function. Then f is said to be
intutionistic fuzzy continuous iff the pre image of each IFS in  is an IFS in  .
Definition 2.9.[9] Let
and
 X , 
(Y , ) be two IFTSs and let f : X  Y be a function. Then f is said to be
intutionistic fuzzy open(resp.closed) iff the image of each IFS in  (resp.(1-  )) is an IFS in  (resp.(1-  )).
An IFTS ( X , T ) represent intutionistic fuzzy topological spaces and for a subset A of a space ( X , T ) , IFcl(A),
Definition 2.10.[8] Let
and
IFint(A), IFPScl(A), IFPSint(A), and A denote an intutionistic fuzzy closure of A, an intutionistic fuzzy interior of A,
intutionistic fuzzy pre semi closure of A, an intutionistic fuzzy pre semi interior of A and the complement of A in X
respectively.
Definition 2.11.[8] A subset A of an IFTS ( X , T ) is called
A  IFcl ( IF int( A)) and an IF semi closed set if IF int( IFcl ( A))  A ;
(ii) An IF  -open if A  IF int( IFcl ( IF int( A))) and an IF  -closed set if IFcl ( IF int( IFcl ( A)))  A ;
(iii) An IF regular open set if IF int( IFcl ( A))  A and an IF regular closed set if IFcl ( IF int( A))  A ;
(iv) An IF pre open set if A  IF int( IFcl ( A)) and an IF pre closed set if IFcl ( IF int( A))  A .
Definition 2.12.[16] A subset A of an IFTS ( X , T ) is called an IF semi pre open set if A  IFcl ( IF int( IFcl ( A))) and
an IF semi pre closed set if IF int( IFcl ( IF int( A)))  A ;
The IF semi closure (resp. IF  closure, IF semi pre closure) of a subset A of ( X , T ) is the intersection of all IF semi
closed (resp. IF  closed, IF semi pre closed) sets that contain A and is denoted by IFs cl ( A) (resp.
IF cl ( A) , IFsp cl ( A) ).
Definition 2.13.[14] A subset A of an IFTS ( X , T ) is called
An IF generalized closed (briefly IF g-closed) set if IF cl ( A)  U whenever A  U and U is IF open in ( X , T ) .
(i) An IF semi open set if
The complement of an IF g-closed set is called an IF g-open set;
Definition 2.14.[13] An IFTS ( X , T ) is called an IF semi T1 if every IF sg-closed set is IF semi closed.
2
Definition 2.15.[13] Let ( X , T ) and
(Y , S ) be IFTSs. A function f : ( X , T )  (Y , S ) is said to be
1
IF semi pre continuous if f (V ) is IF semi pre open in ( X , T ) for every open set V of (Y , S ) .
Notation
IF denotes intuitionistic fuzzy.
IF pre semi open denotes intuitionistic fuzzy pre semi open.
IF pre semi closed denotes intutioinistic fuzzy pre semi closed.
IFPSOS denotes intuitionistic fuzzy pre semi open set.
IFPSCS denotes intuitionistic fuzzy pre semi closed set.
IF regular open denotes intuitionistic fuzzy regular open.
IF  -generalized closed set denotes intuitionistic fuzzy  -generalized closed set.
IF generalized semi pre closed denotes intuitionistic fuzzy generalized semi pre closed.
IF g-open denotes intuitionistic fuzzy IF g-open.
IF  -closed denotes intuitionistic fuzzy  -closed.
IF semi closed denotes intuitionistic fuzzy semi closed.
IF semi open denotes intuitionistic fuzzy semi open.
IF pre closed denotes intuitionistic fuzzy pre closed.
IF gs-closed denotes intuitionistic fuzzy gs-closed.
IF pre semi T 1 space denotes intuitionistic fuzzy pre semi T 1 space.
2
2
IF semi pre
T1 space denotes intuitionistic fuzzy semi pre T1 space.
IF semi pre
T1 space denotes intuitionistic fuzzy semi pre T1 space.
2
3
2
3
IF pre semi continuous function denotes intuitionistic fuzzy pre semi continuous function.
IF pre semi irresolute functions denotes intuitionistic fuzzy pre semi irresolute functions.
3. Properties of IF pre semi closed sets
Definition 3.1. A subset A of an IFTS ( X , T ) is called
(i) IF pre semi closed if IF spcl ( A)  U whenever
A  U and U is IF g-open in ( X , T ) .
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D.Amsaveni, M.K.Uma and E.Roja
(ii) An IF  -generalized closed set (briefly IF g -closed) if IF  cl ( A)  U whenever
A  U and U is IF open in
( X ,T ) ;
(iii) An IF generalized semi pre closed (briefly IF gsp -closed) if IF spcl ( A)  U whenever
A  U and U is IF
open in ( X , T ) .
Definition 3.2. A subset A of an IFTS ( X , T ) is called IF pre semi open if A is IF pre semi closed.
Proposition 3.1. Every IF semi pre closed set is IF pre semi closed set.
Proof. Follows from the fact that IF spcl ( A)  A for any IF semi pre closed set A.
Remark 3.1. The following Example 3.1 shows that the implication in Proposition 3.1 is not reversible.
Example 3.1
Let ( X , T ) be an IFTS. Let
X  a, b and
a b a b
A  x,  ,  ,  ,  ,
 .4 .5   .6 .5 
a b  a b 
B  x,  ,
, ,
 .
 .4 .55   .6 .45 
Then the family
Let
T   0~ ,1~ , A, B  is an IFT on ( X , T ) .
a b  a b 
C  x,  ,
, ,

 .6 .55   .4 .45 
and C is an IF pre semi closed set. But not an IF semi pre closed set. Thus the
class of IF pre semi closed sets properly contains the class of IF semi pre closed sets.
Proposition 3.2. Every IF closed (resp. IF  -closed, IF semi closed, IF pre closed) set is an IF pre semi closed set but the
converses need not be true.
Remark 3.2. The converse of Proposition 3.2 need not be true in general as shown in Example 3.2.
Example 3.2. Follows from the Example 3.1 and the fact that C is an IF pre semi closed set but it is not an IF closed set, IF
 -closed, IF semi closed and IF pre closed.
Proposition 3.3. Every IF pre semi closed set is an IF gsp-closed set.
Proof. The proof is simple.
Remark 3.3. The following Example 3.3 shows that the implication in the Proposition 3.3 is not reversible.
Example 3.3
Let ( X , T ) be an IFTS. Let
X  a, b and
a b a b
A  x,  ,  ,  ,  ,
 .3 .4   .7 .6 
a b a b
B  x,  ,  ,  ,  ,
 .6 .4   .4 .6 
a ba b
C  x,  , ,  , 
 .4 .4   .6 .6 
Then the family
a b a b
T   0~ ,1~ , A, B, C  is an IFT on ( X , T ) .The set F  x,  ,  ,  ,  is IF gsp .3 .5   .7 .5 
closed set, but not an IF pre semi closed set. Therefore every IF gsp-closed set need not be IF pre semi closed.
Remark 3.4. Union of two IF pre semi closed sets need not be IF pre semi closed.
Example 3.4
Let ( X , T ) be an IFTS. Let
X  a, b and
a b a b
A  x,  ,  ,  ,  ,
 .6 .4   .4 .6 
a b a b
B  x,  ,  ,  ,  ,
 .6 .6   .4 .4 
a b a b 
F1  x,  ,  ,  ,  ,
 0 .5   1 .5 
1478
INTUTIONISTIC FUZZY PRE SEMI CLOSED SETS
 a b  a b
F2  x,  ,  ,  ,  .
 .6 0   .4 1 
Then the family T  0~ ,1~ , A, B is an IFT on ( X , T ) . It is easy to verify that F1 and F2 are IF pre semi
closed sets but union of F1 and F2 is not IF pre semi closed set.
Proposition 3.4. If A is IF g-open and IF pre semi closed, then A is IF semi pre closed.
Proof. The proof is simple.
Proposition 3.5. For a subset A of a space ( X , T ) , the following statements are equivalent.
(i)
A is IF open and IF gs-closed.
(ii)
A is IF open and IF gsp-closed.
(iii)
A is IF open and IF pre semi closed.
A is IF regular open.
(iv)
Proof
(i)  (iv): Since A is IF open and every IF open set is IF semi open , IF scl ( A)  A whenever
A  A and A is IF
open in ( X , T ) . But always A  IF scl ( A) , and therefore A  IF scl ( A) which implies that A is IF semi closed. It
follows that
IF int( IFcl ( A))  A . But A  IF int( IFcl ( A)) . Hence A  IF int( IFcl ( A)) .
(iv)  (i): Since A is IF regular open, A is IF semi closed. Therefore IF scl ( A)  A  U where U is IF semi open.
IF int A  IF int( IF cl ( A)) . Since A is IF regular open,
IF int( IF cl ( A))  A and hence IF int( IF cl ( A))  IF int A .Therefore, A  IF int A , implies that A is IF open.
(ii)  (iv): Since A is IF open and gsp-closed, IF spcl ( A)  A , but always A  IF spcl ( A) , and therefore
A  IF spcl ( A) which implies that A is IF semi pre closed. That is, IF int( IFcl ( IF int( A)))  A .Since
A  IF int A , it follows that IF int( IFcl ( A))  A . But A  IF int( IFcl ( A)) . Hence A  IF int( IFcl ( A)) which
And
hence
A
is
IF
gs-closed.
Now,
implies that A is IF regular open.
(iv)  (ii): Since every IF regular open set is IF pre semi closed and every IF pre semi closed set is IF gsp-closed, A is
IF gsp-closed set. Since every g-open set is open, A is open.
(iv)  (iv) follow from the fact that every IF regular open set is IF semi closed, hence IF pre semi closed. (iii)  (ii)
follow from Proposition 3.3.
Proposition 3.6. Let ( X , T ) be an IFTS. If A is an IF pre semi closed set of ( X , T ) , such that A  B  IF spcl ( A) ,
then B is also an IF pre semi closed set of ( X , T ) .
Proof. The proof is simple.
4. Applications of IF Pre Semi Closed Sets
In this section IF pre semi T 1 spaces, IF semi pre
2
T1 spaces are introduced and some applications of IF pre semi closed
3
sets on these spaces are also discussed.
Definition 4.1. IF semi pre T1 space if every IF generalized semi pre closed set is IF semi pre closed.
2
Definition 4.2. An IFTS ( X , T ) is called an IF pre semi
T1 space if every IF pre semi closed set in it is IF semi pre
2
closed.
Proposition 4.1. Every IF semi pre
T1 space is an IF pre semi T1 space.
2
2
Proof. The proof is simple.
Remark 4.1. The converse of Proposition 4.1 need not be true in general as shown in Example 4.1.
Example 4.1
Let ( X , T ) be an IFTS. Let
X  a, b and
a b  a b 
A  x,  ,  ,  ,
 ,
 .4 .31   .6 .69 
a b  a b 
B  x,  ,
, ,
 .
 .4 .45   .6 .55 
Then the family
T  0~ ,1~ , A, B is an IFT on ( X , T ) . Here every IF pre semi closed set in ( X , T ) is IF semi pre
closed set. Therefore ( X , T ) is IF pre semi
T1 . But ( X , T ) is not an
2
1479
IF semi pre
T1 , since
2
D.Amsaveni, M.K.Uma and E.Roja
a b  a b 
C  x,  ,
, ,
 is an IF gsp closed set, but not an IF semi pre closed set. Therefore every IF pre semi
 .4 .35   .6 .65 
T1 space need not be IF semi pre T1 space.
2
2
Definition 4.3. An IFTS ( X , T ) is called an IF semi pre
T1 space if every IF gsp-closed set in it is IF pre semi closed.
3
Proposition 4.2. Every IF semi pre
T1 space is an IF semi pre T1 space.
2
3
Proof. The proof is simple.
Remark 4.2. The converse of the Proposition 4.2 is not true in general as shown in Example 4.2.
Example 4.2. In Example 3.1 ( X , T ) is IF semi pre T 1 space. But not IF semi pre T 1 .
3
2
In Proposition 4.3 it is shown that the dual of the class of IF semi pre
T1 spaces to the class of IF semi pre T1
3
spaces is the class of IF pre semi
2
T1 spaces.
2
Proposition 4.3. An IFTS ( X , T ) is an IF semi pre
T1 iff it is IF semi pre T1 and IF pre semi T1 .
2
3
2
Proof. Necessity: Follows from the proofs of Proposition 3.1 and Proposition 3.3.
Sufficiency: Let A be an IF gsp-closed set of ( X , T ) . Since ( X , T ) is an IF semi pre
T1 space, A is IF pre semi closed.
3
Since ( X , T ) is IF pre semi
T1 space, A is IF semi pre closed set of ( X , T ) . Therefore ( X , T ) is an IF semi pre T1
2
2
space.
5. IF Pre Semi Continuous Functions and IF Pre Semi - Irresolute Functions
Definition 5.1. A function f : ( X , T )  (Y , S ) is called IF pre semi continuous if
f 1 (V ) is an IF pre semi closed set
of ( X , T ) for every IF closed set V of (Y , S ) .
Proposition 5.1. Every IF semi pre continuous function is IF pre semi continuous.
Proof. Follows from Proposition 3.1.
Remark 5.1. The converse of the Proposition 3.1 is not true as as shown in Example 5.1.
Example 5.1
Let ( X , T ) and
(Y , S ) be an IFTSs. Let X  a, b and
a b a b
A  x,  ,  ,  ,  ,
 .4 .5   .6 .5 
a b  a b 
B  x,  ,
, ,
 ,
 .4 .55   .6 .45 
a b  a b 
C  y,  ,
, ,
 .
 .4 .45   .6 .55 
Then the family T   0~ ,1~ , A, B  and S   0~, 1~, C  are IFTs on ( X , T ) and (Y , S ) respectively. Let
f : ( X , T )  (Y , S ) be an identify function.
Now,
a b  a b 
1
f 1 (C )  y,  ,
,  ,
 is an IF pre semi closed set of in ( X , T ) , but f (C ) is
 .6 .55   .4 .45 
not an IF semi pre closed. Hence f is not IF semi pre continuous. However f is an IF pre semi continuous. Therefore
every IF pre semi continuous need not be IF semi pre continuous.
Definition 5.2. Let ( X , T ) and (Y , S ) be an IFTS. A function f : ( X , T )  (Y , S ) is called
IF gsp1
continuous if f (V ) is IF gsp-closed in ( X , T ) for every closed set V of (Y , S ) .
Proposition 5.2. Every IF pre semi continuous function is an IF gsp-continuous function.
Proof. Follows from the fact that every IF pre semi closed set is IF gsp-closed set.
Remark 5.2.The converse of Proposition 5.2 need not be true in general as shown in Example 5.2.
Example 5.2
Let ( X , T ) and
(Y , S ) be an IFTSs. Let X  a, b and
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INTUTIONISTIC FUZZY PRE SEMI CLOSED SETS
a b a b
A  x,  ,  ,  ,  ,
 .3 .4   .7 .6 
a b a b
B  x,  ,  ,  ,  ,
 .6 .4   .4 .6 
a b a b
C  x,  ,  ,  ,  ,
 .4 .4   .6 .6 
a b a b
F  y,  ,  ,  ,  .
 .7 .5   .3 .5 
Then the family
T   0~, 1~, A, B, C  and S   0~, 1~, F  are IFTs on ( X , T ) and (Y , S ) respectively. Let
f : ( X , T )  (Y , S ) be the identify function.
Now,
a b a b
f 1 ( F )  y,  ,  ,  ,  is an IF gsp-closed set, but it is not IF pre semi closed in ( X , T ) .
 .3 .5   .7 .5 
Therefore f is not an IF pre semi continuous. However f is an IF gsp-continuous. Hence every IF gsp-continuous
function need not be IF pre semi continuous function.
Remark 5.3. The composition of two IF pre semi continuous functions need not be IF pre semi continuous function.
Example 5.3
Let ( X , T ) ,
(Y , S ) and ( Z , R) be an IFTSs. Let X   a, b   Y  Z and
a b a b
A  x,  ,  ,  ,  ,
 .4 .5   .6 .5 
a b  a b 
B  x,  ,
, ,
 ,
 .4 .55   .6 .45 
a b a b
C  y,  ,  ,  ,  ,
 .6 .6   .4 .4 
a b  a b 
D  z,  ,
, ,
 .
 .6 .44   .4 .56 
Then the family
T   0~ ,1~ , A, B  , S   0~, 1~, C  and R   0~ ,1~ , D  are IFTs on ( X , T ) , (Y , S ) ,
( Z , R) respectively. Let  : ( X , T )  (Y , S ) and  : (Y , S )  ( Z , R) be identity functions. Clearly  and  are
IF pre semi continuous functions. But    : ( X , T )  ( Z , R ) is not an IF pre semi continuous function. Since
and
a b  a b 
D  z,  ,
, ,

 .4 .56   .6 .44 
is
an
IF
closed
set
of
( Z , R) ,
but
a b  a b 
(   ) 1 ( D )   1 ( 1 ( D ))  z ,  ,
, ,
 is not an IF pre semi closed set of ( X , T ) . Therefore
 .4 .56   .6 .44 
the composition of two IF pre semi continuous functions need not be IF pre semi continuous function.
Definition 5.3. A function f : ( X , T )  (Y , S ) is called IF pre semi irresolute if f
1
(V ) is an IF pre semi closed set of
( X , T ) for every IF pre semi closed set V of (Y , S ) .
Proposition 5.3. Every IF pre semi irresolute function is IF pre semi continuous function, but not conversely.
Proof. The proof is simple.
Proposition 5.4. Let f : ( X , T )  (Y , S ) and g : (Y , S )  ( Z , R ) be any two functions. Then
g  f : ( X , T )  ( Z , R ) is IF pre semi continuous if f is IF pre semi irresolute and g is IF pre semi continuous.
(ii) g  f : ( X , T )  ( Z , R ) is IF pre semi irresolute if both f and g are IF pre semi irresolute.
(iii) g  f : ( X , T )  ( Z , R ) is IF pre semi continuous if f is IF pre semi irresolute and g is IF continuous.
(i)
Proof. The proof is simple.
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D.Amsaveni, M.K.Uma and E.Roja
Proposition 5.5. Let f : ( X , T )  (Y , S ) be an IF pre semi continuous function. If ( X , T ) is IF pre semi
T1 space,
2
then f is IF semi pre continuous.
Proof. The proof is simple.
Proposition 5.6. Let f : ( X , T )  (Y , S ) be an IF gsp-continuous function. If ( X , T ) is an IF semi pre
T1 space
3
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
then f is IF pre semi continuous.
Proof. The proof is simple.
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