Uncertainty Measure and Reduction iUncertainty Measure and Reduction in Intuitionistic Fuzzy Covering Apprn Intuitionistic Fuzzy Covering Appr

oximation Spaceoximation Space

Feng Tao Mi Ju-Sheng

• Basic definitions of IF covering • Uncertainty measure of IFSs in an induced IF coverin

g approximation space• Reduction of IF covering based approximation space• Knowledge Reduction of IF Covering Decision Syste

m based on entropy • Conclusions

where the functions and denote the degree of membership(namely ) and the degree of nonmembership

(namely ) of each element to the set , respectively, and

for each . The family of all IF subsets of is

denoted by .

1. Basic definitions of IF covering1. Basic definitions of IF covering

Definition 1 Let be a nonempty and finite fixed set. denotes the family of all crisp subsets of . An IFS is an object having the form

U ( )P UU

A

A

, ( ), ( ) :A AA x x x x U

: [0,1]A U : [0,1]A U ( )A x

( )A x x U0 ( ) ( ) 1A Ax x x U U

( )IF U

Let is the complement set of . is a constant IFS.A A , { , , : }x x U

Definition 2 Let be a nonempty and finite set called the universe

of discourse. If a family of IFSs

satisfies the conditions that and ,

, then is called an IF covering of .

If , is an IFS, and , then we call covers .

U

, ( ), ( ) : , ( )C Ci ix x x x U i J IF U C

, ( ), ( ) : 1,0i J C Ci ix x x x U i J

, ( ), ( ) : 0,1C Ci ix x x x U C U

x U C ( ) 1C x ( ) 0C x C x

2. Uncertainty measure of IFSs in an induce2. Uncertainty measure of IFSs in an induced IF covering approximation spaced IF covering approximation space

Definition 3 Suppose is a finite and nonempty universe of

discourse, and is an IF covering of . For every ,

let

,

then is another IF covering of , which is called the induced IF covering of based on .

And for every , if , then . Thus if

and , that is, .

,x y U ( ) 1, ( ) 0x xy y C C x yC C

( ) 1, ( ) 0x xy y C C ( ) 1, ( ) 0

y yx x C C x yC C

is the set of all most basic granules and we use to define the IF covering upper and lower approximation operators.{ : }x x UC { : }x x UC

1{ , , }nC C C U

U

x U

{ : , ( ) 1, ( ) 0}x j j C Cj jC C x x C C

( ) { : }xCov x U C CU

UC

Definition 4 Let be a finite and nonempty universe of discourse, an IF covering of . For any , the induced IF covering upper and lower approximations of w.r.t. , denoted by and , are two IFSs and are defined, respectively, as follows: , .Where , ;

, . are referred to as induced IF covering upper and lower rough approximation operators w.r.t. , respectively. Thepair is called an induced IF covering approximation space. , if , then is called inner definable; if , then is outer definable. is definable iff .

U

C U ( )A IF UA ( )Cov C ( )AC ( )AC

( ) ( )( ) , ( ), ( ) :

A AA x x x x U

C CC ( ) ( )( ) , ( ), ( ) :A AA x x x x U C CC

( )( ) [ ( ) ( )]y U AA xx y y

C C ( )( ) [ ( ) ( )]y U AA xx y y CC

( ) ( ) [ ( ) ( )]A y U Axx y y C C ( ) ( ) [ ( ) ( )]A y U Ax

x y y C C

, : ( ) ( )IF U IF UC C

( )Cov C

( , ( ))U Cov C

( )A IF U ( )A AC A ( )A AC

A

A

( ) ( )A AC C

Theorem 1 Let be an induced IF covering approximation

space. Then the induced IF covering upper and lower rough approximation

operators w.r.t. satisfy the following properties: ,

with ,

( , ( ))U Cov C

( )Cov C , ( )A B IF U , [0,1] 1

(1) ( ) ( ), ( ) ( );A A A A C C C C

(2) ( , ) ( ) , , ( , ) ( ) , ;A A A A C C C C

(3) ( ) ( ) ( ), ( ) ( ) ( );A B A B A B A B C C C C C C

(4) ( ) ( ), ( ) ( );A B A B A B C C C C

(5) ( ) , ( ) ;U U C C

(6) ( ) ( ).A A A C C

( , ( ))U Cov C

( )A IF U

Theorem 2 Suppose is an induced IF covering

approximation space, and .

(1)If is inner definable, then , and

.

A ,x y U ( ) ( ) ( )A Axy y x

C

( ) ( ) ( )A Axy y x C

(2) If is outer definable, then , and

.

A ,x y U ( ) ( ) ( )A Axy y x

C

( ) ( ) ( )A Axy y x C

(3) If is definable, then , and

.

A ,x z U ( ) ( ) ( ( ) ( ))A y U Ax xz z y y C C

( ) ( ) ( ( ) ( ))A y U Ax xz z y y C C

Theorem 3 Suppose is an induced IF covering

approximation space. Then is definable iff

and whenever and .

( , ( ))U Cov C

A U ( ) 0 ( ) 1xx

y y CC

( ) 0 ( ) 1yy

x x CC x A y A

Definition 5 Let be a given finite and nonempty universe of discourse, . A real function is referred to as an entropy on

, if it satisfies the following properties:(1) iff ;(2) iff , ;(3) When , , if , , or when , if

and , then ;(4) , is called a intuitionistic fuzzy entropy (IFE, for short) of .For each IFS , let , then E(A) is an entropy on .

U

( )A IF U : ( ) [0,1]E IF U ( )IF U

( ) 0E A ( )A P U( ) 1E A ( ) ( )A Ax x x U

x U ( ) ( )A Ax x * ( ) ( )AAx x * ( ) ( )AA

x x

( ) ( )A Ax x * ( ) ( )AAx x * ( ) ( )AA

x x *( ) ( )E A E A

( ) ( )E A E A ( )E A A

( )A IF U2 2

2 2

1 ( ( ) ( )) (1 ( ) ( ))1( )

| | 1 ( ( ) ( )) (1 ( ) ( ))A A A A

x U A A A A

x x x xE A

U x x x x

( )IF U

Definition 6 Let be an IF covering of , can be

considered as the outcomes of an experiment . The information gained

by performing the experiment is the expectation

. (1)

Where denotes the element number of . is referred to as the

entropy of the IF covering .

1{ , , }nC C C U 1, , nC C

C

C

2 2

2 2 2

1 ( ( ) ( )) (1 ( ) ( ))1

| | 1 ( ( ) ( )) (1 ( ) ( ))x x x x

x x x xx U y U

y y y yE

U y y y y

C C C C

C C C C

C

| |U U E C

C

Let and be two IF coverings of , , we defineB C U y U

0, ( ) ( )

( ) ( )( ),

2 |{ : ( ) 1} |

x x

x xx x

x

y y

y yyotherwise

y y

C B

C BC B

C

1, ( ) ( ) ( ) ( )

( ) ( )( ) , ( ) ( ) ( ) ( )

2 |{ : ( ) 1} |

( ) ( ),

2 |{ : ( ) 1} |

x x x x

x x

x x x x x x

x

x x

x

y y and y y

y yy y y and y y

y y

y yotherwise

y y

C B C B

C BC B C B C B

C

B C

C

then, is an IFS.x xC B

Definition 7 Let and be two IF coverings of . The

conditional entropy of given is defined by

. (2)

measures the uncertainty about the outcomes of the

experiment associated with the IF covering given the outcomes

of the experiment represented by .

B C

UB C

2 2

2 2 2

1 ( ( ) ( )) (1 ( ) ( ))1|

2 | | 1 ( ( ) ( )) (1 ( ) ( ))x x x x x x x x

x x x xx U y U

y y y yE

U y y y y

C B C B C B C B

C C C C

B C

|E B C

BC

Proposition 1 Suppose be a finite and nonempty universe, and are two IF coverings of . ThenUCB

U

(1) 0 ( ) 1;E C

(2) 0 | 1;E B C

(3) , | 0.If then E C B B C

Definition 8 Let be a finite and nonempty universe, an IF

covering of , for every , , define as:

,

Where , .

is called the degree of membership of w.r.t. based on an IF covering

. is called the degree of non-membership of

w.r.t. based on an IF covering . is still a IFS.

U C

U ( )A IF U ( ) : ( ) ( )R A IF U IF UCx U

( ) ( ) min{ ( ), ( )};R A y U Axx y y CC

( ) ( ) max{ ( ), ( )}.R A Axy Ux y y

CC

a b a b ab , [0,1]a b

( ) ( )R A xC

x

x

A

A

( )Cov C

( )Cov C ( ) ( )R A xC

( )R AC

( ) ( )( ) ( )R A R Ax x B C(1) If is finer than , then and ,B C ( ) ( )( ) ( )R A R Ax x

B C

Proposition 2 Let be a finite and nonempty universe,

two IF covering of . , , ( )A B IF U U

,B CU

x U

( ) ( )A Bx x (2) If and , then and

( ) ( )A Bx x ( ) ( )( ) ( )R A R Bx x C C

( ) ( )( ) ( ),R A R Bx x C C

(3) , , ( ) ( ) ( )R A B R A R B C C C ( ) ( ) ( )R A B R A R B C C C

( ) ( ) ( )R A B R A R B C C C(4) If or , , ,( ) ( ) ( )R A B R A R B C C CA B B A

(5) , .( )A P U ( ) ( ) 1,0R A R A C C

Where

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) , ( ) ( ) ( ) ( ), ( ) ( ) :R A R A R A R A R A R AR A R A x x x x x x x x U C C C C C C C C

Definition 9 Suppose is a finite nonempty universe of discourse, is a IF covering of , is the cardinality of , , define

, (3)

then is called the IF entropy of with respect to .

( )A IF UUnUCU

2 2( ) ( ) ( ) ( )

2 2( ) ( ) ( ) ( )

1 ( ( ) ( )) (1 ( ) ( ))1( )

1 ( ( ) ( )) (1 ( ) ( ))R A R A R A R A

x U R A R A R A R A

x x x xIFR A

n x x x x

( )IFR A A ( )Cov C

Proposition 3 Let be an IF covering of a universe of

discourse , ,( )A P U U

C

2 2

2 2

1 ( ( ) ( )) (1 ( ) ( ))1(1) ( ) ,

1 ( ( ) ( )) (1 ( ) ( ))

y A y Ax x x xy A y A

x A y A y Ax x x xy A y A

y y y yIFR A

n y y y y

C C C C

C C C C

2 2

2 2

1 ( ( ) ( )) (1 ( ) ( ))1(2) ( ) .

1 ( ( ) ( )) (1 ( ) ( ))

y A y Ax x x xy A y A

x A y A y Ax x x xy A y A

y y y yIFR A

n y y y y

C C C C

C C C C

Theorem 4 Suppose is a finite and nonempty universe of

discourse, is an IF covering of . , if and is

definable, then .

U

C U ( )A IF U ( )A P U A

( ) 0IFR A

3. Reduction of IF covering based approximation space

Definition 10 Suppose is a finite and nonempty universe of

discourse. Let be an IF covering of , and . If is a union

of some IFSs in , then is called a reducible element of ,

otherwise is an irreducible element of . And if every element of

is an irreducible element, we say that is irreducible; otherwise

is reducible.( Similarly to Prof. Zhu)C

CC

C

C{ }KC

KK CUC

U

K

Proposition 4 Let be an IF covering of .

(1) If is a reducible element of , then is also an IF covering

of .

(2) is a reducible element of , and , then is a

reducible element of if and only if is a reducible element

of .{ }KC1K

C 1K1 { }K K C

CK

UC

{ }KC

U

KC

K

Definition 11 For an IF covering of a universe , the new

irreducible IF covering through deleting all reducible elements is

called the reduction of IF covering , denoted by .Re ( )d CC

UC

Proposition 5 Let be a finite and nonempty universe of

discourse, an IF covering of . If be a reducible element of ,then

U

C U KC

( )x xK C C(1) , for every .x U

(2) , for every .Re ( )x xdC C x U

(3) , .( ) Re ( )( )X d XC C ( )X IF U

(4) , . ( ) Re ( )( )X d XC C ( )X IF U

Definition 12 Let be a finite and nonempty universe of

discourse, a finite IF covering of , and

, then is a superfluous element of . If is

an IF covering of satisfying and and none of

element in is superfluous, then is called the IF approximation

reduction of , denoted by .Re( )C

BB

R RC BR RC BU

B CCiC( )CiR R C C

( )CiR R C C,iC CUC

U

C

Theorem 5 Let be a finite and nonempty universe of disco-

urse, and a finite IF covering of . Then is a superfluous element

of iff , for all . x U( )x xK C CC

UC

U

iC

Proposition 6 Let be a finite and nonempty universe of dis-

course, a finite IF covering of , and be a reducible element

of , then is a superfluous element of .iCCiCUC

U

C

Proposition 7 Suppose is a finite IF covering of ,

is a superfluous element, then satisfies one of the following

condition:

C U iC C

iC

( ) 1Cix (1) or , .( ) 0Ci

x x U

(2) , , satisfies ,

and .

{ : ( ) 1, ( ) 0}C Ci ix y y y iC S C ( ) 1x S

( ) 0x S iC S

Algorithm 1 Suppose is a finite and nonempty set,

is a finite IF covering of .U1{ , , }rC C CU

Step 1. For every

compute and .

end for;

Let .

Step 2. For i from 1 to r

if or ,

then ;

end if;

end for;

Step 3. Output S.

x U* { : ( ) 1, ( ) 0}i C Ci ix C x x C ** *x x

S C

iC S S

An algorithm for computing a reduction of an IF covering based approximation space

{ : ( ) 1, ( ) 0}C Ci iy U y y ** *( )ix x C

4. Knowledge Reduction of Consistent IF 4. Knowledge Reduction of Consistent IF Covering Decision System based on entropy Covering Decision System based on entropy

In an induced IF covering approximation space , ,the positive region of w.r.t. is computed by the following formula: .If is an IF covering as an IF decision, then is an IF covering decision system. Hence

( , ( ))U Cov C ( )X IF U

X ( )Cov C

( ) { : }xPos X x U X C

D ( , ( ) ( ))U Cov Cov DC

( ) { : }.x xPos D x U D C C

If for every , , then decision system

is called a consistent IF covering decision system.

x U x xDC ( , ( ) ( ))U Cov Cov DC

In this section, we let .1,0

Definition 13 Let be a consistent IF covering

decision system. For , if for all , , then is

called dispensable about in , otherwise is called indispensable.

For every IF subcovering , if , , then is a

consistent set w.r.t. , if every element in is

indispensable, i.e., for every , , , then is called an

independent about . If is consistent and independent subset, then

is a reduction of about . The collection of all the indispensable

elements in is called the core of about , denoted by .

( , ( ) ( ))U Cov Cov DC

iC C x U ( )i x xC D C iC

D C iC

B C x xDB x U B( , ( ) ( ))U Cov Cov DC B

iC B x xDB x U B

D

B

B

C D

CC D ( )DCore C

For every IF subcovering , the conditional entropy of to

is B C D B

2 2

2 2 2

1 ( ( ) ( )) (1 ( ) ( ))1|

2 | | 1 ( ( ) ( )) (1 ( ) ( ))x x x x x x x x

x x x x

D D D D

x U y U

y y y yE D

U y y y y

B B B B

B B B B

B

Theorem 6 Let be a consistent IF covering

decision system. For every , is dispensable about in iff

.

( , ( ) ( ))U Cov Cov DC

iC C iC D C

| | iE D E D C C C

Theorem 7 Let be a consistent IF covering

decision system and . is a reduction about in if and

only if

( , ( ) ( ))U Cov Cov DC

B C B D C

(1) ; | |E D E DB C

(2) is independent about .B D

Definition 14 Let be a consistent IF cov-ering decision system. For every , is a IF subcovering. We define the significance of the IFS w.r.t. in by

( , ( ) ( ))U Cov Cov DCiC B C B

iC D B

( ) | | | | .i iSig C E D C E D B B B

Theorem 8 Let be a consistent IF covering

decision system. For every , is indispensable about in

if and only if .

( , ( ) ( ))U Cov Cov DC

iC C iC D C

( ) 0iSig C C

Theorem 9 .( ) { : ( ) 0}D i iCore C Sig C CC

Proposition 8 Let be a consistent IF cov-

ering decision system, , , if , then ,

.

( , ( ) ( ))U Cov Cov DC

x U B C x xDB iC C B

( )i x xC DB

Algorithm 2 Reduction of a consistence IF covering decision system

Step 1. Let ;

Step 2. For each calculate ;

if , then . end if ;

end for ;

Let .

Step 3. While ,

for each calculate ; end for;

select , such that ,

let .

end while;

Step 4. Output .

( )DCore C

iC C ( )iSig CC

( ) 0iSig C C( ) ( )D D iCore Core C C C

( )DCoreB C

C C B ( )CSig CB

( ) max{ ( ) : }iC C i iSig C Sig C C B B C B

C B B

B

| |E D E DB C

C

5.Conclusions5.Conclusions

In this paper, we proposed a pair of new definitions of induced IF

upper and lower approximation based on an IF covering and disc

ussed their properties. Then, we defined an uncertainty measure

of IF sets in an induced IF covering approximation space. The red

uction of an IF covering was also studied. Finally, we discuss the

reduction of IF covering decision systems using condition entropy.

In the future, we will pay more attention to the study of uncertaint

y in IF covering environments.

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