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DOI: 10.1142/S0218488513500141

International Journal of Uncertainty,Fuzziness and Knowledge-Based SystemsVol. 21, No. 2 (2013) 263276c World Scientific Publishing Company

SOME OPERATIONS OVER ATANASSOVS INTUITIONISTIC

FUZZY SETS BASED ON EINSTEIN T-NORM AND T-CONORM

WEIZE WANG and XINWANG LIU

School of Economics and Management, Southeast University,

Nanjing, 211189, P.R. [email protected]@seu.edu.cn

Received 17 November 2010Revised 15 November 2012

We define some operations over Atanassovs intuitionistic fuzzy sets (AIFSs), such asEinstein intersection, Einstein product, Einstein scalar multiplication and Einstein ex-

ponentiation, etc., and then define new concentration and dilation of AIFSs. Thesedefinitions will be useful while dealing with various linguistic hedges like very, moreor less, highly, very very etc., involved in the problems under intuitionistic fuzzyenvironment. We also prove some propositions and present some examples in this con-text.

Keywords: Atanassovs intuitionistic fuzzy sets (AIFSs); Einstein operation; CON;DIL; linguistic variables.

1. Introduction

Atanassovs Intuitionistic fuzzy set (AIFS),1 a generalization of fuzzy set,2 has beenfound to be highly useful to deal with vagueness. Each element in an AIFS is ex-

pressed by an ordered pair, and each ordered pair is characterized by a membership

degree and a non-membership degree, the sum of the membership degree and the

non-membership degree of each ordered pair is less than or equal to one. Since it

was first introduced by Atanassov in 1986,1 the AIFS theory has been widely inves-

tigated and applied to a variety.37 The operation over AIFSs is an interesting and

important research topic in AIFS theory that has been receiving more and more

attention in recent years.1,817 Atanassov1,8,9 defined some basic operations and

relations of AIFSs, including intersection, union, complement, algebraic sum andalgebraic product, etc., and proved an equality between AIFSs. Atanassov10 de-

termined the pseudo-fixed points of all operators defined over AIFSs. Atanassov11

Corresponding author.

263

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264 W.-Z. Wang & X.-W. Liu

formulated and proved two theorems related to the relations between some of the

operators over AIFSs. De et al.12 further defined the concentration, dilation and

normalization of AIFSs, and proved some propositions. Deschrijver and Kerre13 in-

troduced some aggregation operators on the lattice L, and defined the special classes

of binary aggregation operators, based on triangular norms (t-norms) on the unit

interval (Deschrijver and Kerre1416 showed that AIFSs could be seen as L-fuzzy

sets). However, there is only one type of the operational laws of AIFS, that is to say,

these operations are based on the product t-norm and its dual t-conorm, which are

widely used in decision making. Although Deschrijver and Kerre17 have introduced

a generalised union and a generalised intersection of AIFSs using a general t-norm

and t-conorm, the extensions of the scalar multiplication and exponentiation oper-

ations on AIFS are limited for the structures of various t-norms are different.18 The

Einstein t-norm is the same strict t-norm as the product t-norm, and the Einstein

t-norm and its dual t-conorm have unique structures. Therefore, this paper focuses

mainly on another type of operations on AIFS, which are based on the Einstein

t-norm and its dual t-conorm.

In the present paper, motivated by the generalised union and intersection over

AIFSs,17 we introduce some operations over AIFSs, such as Einstein intersection,

Einstein product. Then we deduce Einstein scalar multiplication and Einstein ex-

ponentiation over AIFSs. Finally, we define new concentrated AIFS, dilated AIFS

and make some characterizations.

2. Preliminaries

In this section, the basic concepts of AIFSs and some existing operations over AIFSs

will be introduced below to facilitate future discussions.

Definition 1. Let a set Ebe fixed, an AIFS A ofE is defined as:

A= {(x, A(x), A(x))|x E}

where the mappings A : E [0, 1] and A : E [0, 1] satisfy the condition 0

A(x) +A(x) 1, x E, and they denote the degrees of membership and

nonmembership of element x Eto set A, respectively.

LetA(x) = 1A(x)A(x), then it is usually called the Atanassovs intuition-

istic fuzzy index of x A, representing the degree of indeterminacy or hesitation

ofxto A. It is obvious that 0 A(x) 1 for every x E .

Atanassov1 described some operations over AIFSs as follows.

Definition 2. IfA and B are two AIFSs of the set E, then

A B iffx E, A(x) B(x) & A(x) B(x),

A= B iffA B & BA,

A= {(x, A(x), A(x)|x E},

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Some Operations Over Atanassovs Intuitionistic Fuzzy Sets 265

A B ={(x, min(A(x), B(x)), max(A(x), B(x)))|x E},

A B ={(x, max(A(x), B(x)), min(A(x), B(x)))|x E}, A B ={(x, A(x)B(x), A(x) + B(x) A(x)B(x))|x E},

A B ={(x, A(x) + B(x) A(x)B(x), A(x)B(x))|x E},

A= {(x, A(x), 1 A(x))|x E},

A= {(x, 1 A(x), A(x))|x E}.

De et al.12 further defined the following operations over AIFSs.

Definition 3. The scalar multiplication operation over an AIFSAof the universe

Eis denoted by nA and is defined by

nA= {(x, 1 [1 A(x)]n, [A(x)]

n)|x E}

where n is any positive real number.

Definition 4. The exponentiation operation over an AIFS A of the universe E is

denoted by An and is defined by

An ={(x, [A(x)]n, 1 [1 A(x)]

n|x E}

where n is any positive real number.

Definition 5. The concentration of an AIFS A of the universe E, is denoted by

CON(A) and is defined by

CON(A) ={(x, CON(A)(x), CON(A)(x))|x E}

where CON(A)(x) = [A(x)]2, CON(A)(x) = 1 [1 A(x)]

2.

In other words, concentration of an AIFS is defined by CON(A) = A2. The

linguistic variable very is viewed as labels of the concentration on AIFSs of the

universe of discourse.12

Definition 6. The dilation of an AIFS Aof the universe E, is denoted by DIL(A)

and is defined by

DIL(A) ={(x, DIL(A)(x), DIL(A)(x))|x E}

where CON(A)(x) = [A(x)]1/2, CON(A)(x) = 1 [1 A(x)]

1/2.

In other words, dilation of an AIFS is defined by CON(A) =A1/2.The linguistic

variable more or less is viewed as labels of the dilation on AIFSs of the universe

of discourse.12The set theoretical operations have had an important role since the beginning

of fuzzy set (FS) theory.2 Starting from Zadehs operations min and max, many

other operations were introduced in the FS literature. All types of the particular

operations were included in the general concepts of t-norms and t-conorms,17,19,20

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which satisfy the requirements of the conjunction and disjunction operators, respec-

tively. Thus, the t-normTand t-conorm Sare the most general families of binary

functions that map the unit square into the unit interval.

Next, we introduce some examples of t-norms and t-conorms,17,19,20 which sat-

isfy the requirements of the conjunction and disjunction operators, respectively.

Zadeh-intersection TMis a t-norm, Zadeh-union SM is a t-conorm, where

TM(a, b) = min{a, b}, SM(a, b) = max{a, b} . (1)

Algebraic product TP is a t-norm and Algebraic sum SP is a t-conorm, where

TP

(a, b) =a b, SP

(a, b) =a + b a b . (2)

Einstein product T is a t-norm and Einstein sum S is a t-conorm, where

T(a, b) = a b

1 + (1 a) (1 b), S(a, b) =

a + b

1 + a b (3)

where each pair (T , S) is related by the De Morgan duality: S(a, b) = 1 (T(1

a, 1 b)), (a, b) [0, 1]2.

Based on the general t-norm and t-conorm, Deschrijver and Kerre17 proposed

the generalised intersection and union of AIFSs A and B, respectively, as follows:

AT,SB={(x, T(A, B), S(A, B))|x E} (4)

AS,TB={(x, S(A, B), T(A, B))|x E} (5)

where any pair (T, S) can be used, Tdenotes a t-norm and Sa so-called t-conorm

dual to the t-norm T, defined by S(x, y) = 1 T(1 x, 1 y). Furthermore, the

generalised intersection (AT,SB) and union (AS,TB) are AIFSs ofE.

For instance, the Atanassov-intersection A B and the Atanassov-union A B

of two AIFSs A and B can be obtained by the t-norm TM and the t-conorm SM;

the algebraic product A B and the algebraic sum A B by the algebraic product

TPand the algebraic sum SP, i.e.,

A B={(x, TM(A(x), B(x)), SM(A(x), B(x)))|x E},

A B={(x, SM(A(x), B(x)), TM(A(x), B(x)))|x E},

A B={(x, TP(A(x), B(x)), SP(A(x), B(x)))|x E},

A B={(x, SP(A(x), B(x)), TP(A(x), B(x)))|x E}.

3. Some Operations over AIFS Based on Einstein t-norm and

t-conormThe set theoretical operators have had an important role since in the beginning of

AIFS theory. In this section, let the t-normTand its dual t-conorm Sbe Einstein

product T and Einstein sum S, respectively, then the generalised intersection (4)

and union (5) over two AIFSs A and Bbecome Einstein product (denoted by AB)

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and Einstein sum (denoted by AB) over two AIFSs A and B, respectively, as

follows:

AB={(x, T(A(x), B(x)), S(A(x), B(x)))|x E}, (6)

AB ={(x, S(A(x), B(x)), T(A(x), B(x)))|x E}. (7)

Based on Einstein t-norm and its dual t-conorm (3), Einstein product (6) and

Einstein sum (7) over two AIFSs A and B are further indicated as the following

operations:

AB= x,

A(x) B(x)

1 + (1 A(x)) (1 B(x)),

A(x) + B(x)

1 + A(x) B(x)|x E

(8)

AB=

x,

A(x) + B(x)

1 + A(x) B(x),

A(x) B(x)

1 + (1 A(x)) (1 B(x))

|x E

(9)

Theorem 1. If n is any positive integer and A is an AIFS of E, then the expo-

nentiation operation is a mapping fromZ+ EtoE:

An =

x,

2[A(x)]n

[2 A(x)]n

+ [A(x)]n ,

[1 + A(x)]n [1 A(x)]

n

[1 + A(x)]n

+ [1 A(x)]n

|x E

(10)

whereAn =

n

AA . . . A .Proof. Mathematical induction can be used to prove that the above Eq. (10) holds

for all positive integers n. The Eq. (10) is called P(n).

Basis: Show that the statement P(n) holds for n= 1, and the statement P(n)

amounts to the statement P(1),i.e.,

A1 =

x,

2[A(x)]

[2 A(x)] + [A(x)],[1 + A(x)] [1 A(x)]

[1 + A(x)] + [1 A(x)]

|x E

In the left-hand side of the equation, A

1

=A = {(x, A(x), A(x))|x E}.In the right-hand side of the equation,x,

2[A(x)]

[2 A(x)] + [A(x)],[1 + A(x)] [1 A(x)]

[1 + A(x)] + [1 A(x)]

|x E

={(x, A(x), A(x))|x E} .

The two sides are equal, so the statement P(n) is true for n = 1. Thus it has

been shown the statement P(1) holds.

Inductive step: Show that ifP(n) holds, then also P(n + 1) holds.

Assume P(n) holds (for some unspecified value of n). It must then be shown

that P(n + 1) holds, that is:

A(n+1)

=

x,

2[A(x)]n+1

[2 A(x)]n+1

+ [A(x)]n+1 ,

[1 + A(x)]n+1 [1 A(x)]

n+1

[1 + A(x)]n+1

+ [1 A(x)]n+1

|x E

.

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Using the induction hypothesis that P(n) holds, the left-hand side can be

rewritten to:

A(n+1) =AnA ,

and based on the Einstein sum operation over two AIFSs, we have

AnA

=

x,

2[A(x)]n+1

[2 A(x)]n+1

+ [A(x)]n+1 ,

[1 + A(x)]n+1 [1 A(x)]

n+1

[1 + A(x)]n+1

+ [1 A(x)]n+1

|x E

,

thereby showing that indeed P(n + 1) holds.

Since both the basis and the inductive step have been proved, it has now beenproved by mathematical induction that P(n) holds for any positive integer n.

Next, we prove the result ofAn is also an AIFS even ifnis any positive real

number, i.e.,

0 2[A(x)]

n

[2 A(x)]n

+ [A(x)]n +

[1 + A(x)]n

[1 A(x)]n

[1 + A(x)]n

+ [1 A(x)]n 1 .

The left-hand side of the inequality holds obviously, we prove the right-hand

side of the inequality.

Since 0 A(x) 1, 0 A(x) 1, 0 A(x) + A(x) 1, then 1 A(x) A(x) 0, thus

2[A(x)]n

[2 A(x)]n

+ [A(x)]n =

2[A(x)]n

[1 + (1 A(x))]n

+ [A(x)]n

2[A(x)]n

[1 + A(x)]n

+ [A(x)]n .

Also since 1 A(x) A(x) 0, then

[1 + A(x)]n [1 A(x)]

n

[1 + A(x)]n + [1 A(x)]

n [1 + A(x)]

n [1 A(x)]n

[1 + A(x)]n + [A(x)]

n [1 + A(x)]

n [A(x)]n

[1 + A(x)]n + [A(x)]

n .

Thus

2[A(x)]n

[2 A(x)]n

+ [A(x)]n +

[1 + A(x)]n [1 A(x)]

n

[1 + A(x)]n

+ [1 A(x)]n

2[A(x)]

n

[1 + A(x)]n

+ [A(x)]n +

[1 + A(x)]n [A(x)]

n

[1 + A(x)]n

+ [A(x)]n = 1.

That is to say, the AIFS An defined above is an AIFS.

Similarly, we can get new scalar multiplication operation over AIFSs as follows:

Theorem 2. If n is any positive number andA is an AIFS ofE, then the scalar

multiplication operation is a mapping from R+ Eto E

nA=

x,

[1 +A(x)]n

[1 A(x)]n

[1 +A(x)]n

+ [1 A(x)]n ,

2[A(x)]n

[2 A(x)]n

+ [A(x)]n

|x E

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where nA=

n

AA . . . A, and the result of nA is also an AIFS even if nis any positive real number.

Example 1. Consider the universe E={1, 2, 3, 4, 5}, and let Abe an AIFS ofE

given by A= {(0.2, 0.6), (0.3, 0.6), (0.5, 0.5), (0.6, 0.3), (0.9, 0)}.

Then we see that

A2 ={(0.0244, 0.8824), (0.0604, 0.8824), (0.2, 0.8), (0.3103, 0.5505), (0.802, 0)},

A0.5 ={(0.5, 0.3333), (0.5916, 0.3333), (0.7321, 0.2679), (0.7913, 0.1535), (0.9499, 0)},

2A= {(0.3846, 0.3103), (0.5505, 0.3103), (0.8, 0.2), (0.8824, 0.0604), (0.9945, 0)},

0.5A= {(0.101, 0.7913), (0.1535, 0.7913), (0.2679, 0.7321), (0.3333, 0.5916),

(0.6268, 0)}.

Proposition 1. For an AIFS A of the setEand for any positive numbern.

(i) An = (A)n,

(ii) An = (A)n,

(iii) IfA(x) = 0, thenAn(x) = 0

(iv) Am An, where m and n are both positive numbers andm n.

(v) IfA is totally intuitionistic, then An is also so.

Proof.

(i) An =

x,

2[A(x)]n

[2 A(x)]n + [A(x)]

n

|x E

= (A)n

(ii) An =

x,

2[1 A(x)]n

[1 + A(x)]n

+ [1 A(x)]n

|x E

= (A)n

(iii) A(x) = 0 A(x) + A(x) = 1, then

An =

x,

2[A(x)]n

[2 A(x)]n + [A(x)]

n ,[1 + A(x)]

n [1 A(x)]n

[1 + A(x)]n + [1 A(x)]

n

|x E

=

x,

2[A(x)]n

[2 A(x)]n

+ [A(x)]n ,

[2 A(x)]n [A(x)]

n

[2 A(x)]n

+ [A(x)]n

|x E

,

which gives An(x) = 0.

(iv) Since

An

= x, 2[A(x)]n

[2 A(x)]n + [A(x)]n ,[1 + A(x)]

n [1 A(x)]n

[1 + A(x)]n + [1 A(x)]n |x E

and

Am =

x,

2[A(x)]m

[2 A(x)]m

+ [A(x)]m ,

[1 + A(x)]m

[1 A(x)]m

[1 + A(x)]m

+ [1 A(x)]m

|x E

.

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Let

f1(x) = 2x

(2 )x

+ x, g1(y) = (1 + )

y

(1 )

y

(1 + )y

+ (1 )y (11)

then we have

f1(x) =2x(2 )x ln

2

[(2 )x + x]2 0 (12)

g1(y) =2(1 + )

y(1 )

y[ln(1 + ) ln(1 )]

[(1 + )y

+ (1 )y

]2 >0, (0, 1), x >0 (13)

which show that f1(x) and g1(y) are decreasing and increasing functionsrespectively. So if m n, then f1(m) f1(n) and g1(m) g1(n),

i.e.,Am(x) An(x) and Am(x) An(x), thus Am An.

Hence proved.

(v) Straightforward.

Proposition 2. For an AIFSA of the setEand for any positive numbern

(i) (nA) =n(A),

(ii) (nA) =nA,(iii) IfA(x) = 0,,thennA(x) = 0,

(iv) mA nA, wheremandnare both positive numbers andm n,

(v) IfA is totally intuitionistic, thennA is also so.

Proof. Similar to Proposition 2.

Proposition 3. Consider two AIFSs A and B of the set E, for any positive

numbern,

(i) IfA B, thenAn Bn,

(ii) IfA B, thennA nB,

(iii) (A B)n =An Bn,

(iv) (A B)n =An Bn,

(v) n(A B) =nA nB,

(vi) n(A B) =nA nB.

Proof.

(i) Since

An =

x,

2[A(x)]n

[2 A(x)]n

+ [A(x)]n ,

[1 + A(x)]n

[1 A(x)]n

[1 + A(x)]n

+ [1 A(x)]n

|x E

.

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Let

f2(x) = 2xn

(2 x)n + xn , g2(y) =

(1 + y)n (1 y)n

(1 + y)n

+ (1 y)n (14)

then we have

f2(x) = 4nxn1(2 x)

n1

[(2 x)n

+ xn]2 >0, x (0, 1) (15)

and

g2(y) =4n(1 + y)n1(1 y)n1

[(1 + y)n

+ (1 y)n

]2 >0, y(0, 1) . (16)

It show that both f2(x) and g2(y) are increasing functions. So ifA B, i.e.,x E, A(x) B(x) & A(x) B(x), then f2(A(x)) f2(B(x)) and

g2(A(x)) g2(B(x)), i.e., An(x) Bn(x) and An(x) Bn(x)

thus An Bn,

(ii) Similar to (i),

(iii) Since A B={(x, min{A(x), B(x)}, max{A(x), B(x)})|x E}, then

(A B)n ={(x, (AB)n(x), (AB)n(x))|x E}

where

(AB)n(x) = 2[min{A(x), B(x)}]n

[2 min{A(x), B(x)}]n

+ [min{A(x), B(x)}]n ,

and

(AB)n(x) =[1 + max{A(x), B(x)}]

n [1 max{A(x), B(x)}]n

[1 + max{A(x), B(x)}]n + [1 max{A(x), B(x)}]

n .

Also since the (11)(13) hold, i.e., f2(x) andg2(y) are increasing functions, we

have

(AB)

n(x) =

2[min(A(x), B(x))]n

[2 min(A(x), B(x))]n + [min(A(x), B(x))]n

= min

2[A(x)]

n

[2 A(x)]n

+ [A(x)]n ,

2[B(x)]n

[2 B(x)]n

+ [B(x)]n

=AnBn(x)

and

(AB)n(x) =[1 + max{A(x), B(x)}]

n [1 max{A(x), B(x)}]n

[1 + max{A(x), B(x)}]n

+ [1 max{A(x), B(x)}]n

= max [1 + A(x)]n [1 A(x)]n

[1 + A(x)]n

+ [1 A(x)]n ,

[1 + B(x)]n [1 B(x)]n

[1 + B(x)]n

+ [1 B(x)]n

=AnBn(x) .

That is to say (A B)n =An Bn.

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(iv) Since A B={(x, max{A(x), B(x)}, min{A(x), B(x)})|x E} then

(A B)n ={(x, (AB)n(x), (AB)n(x))|x E}

where

(AB)n(x) = 2[max{A(x), B(x)}]

n

[2 max{A(x), B(x)}]n

+ [max{A(x), B(x)}]n

and

(AB)n(x) =[1 + min{A(x), B(x)}]

n [1 min{A(x), B(x)}]n

[1 + min{A(x), B(x)}]n

+ [1 min{A(x), B(x)}]n .

Also since the (14)(16) hold, i.e., f2(x) andg2(y) are increasing functions, we

have

(AB)n(x) = 2[max{A(x), B(x)}]

n

[2 max{A(x), B(x)}]n + [max{A(x), B(x)}]

n

= max

2[A(x)]

n

[2 A(x)]n

+ [A(x)]n ,

2[B(x)]n

[2 B(x)]n

+ [B(x)]n

= AnBn(x) .

and

(AB)n(x) =[1 + min{A(x), B(x)}]

n [1 min{A(x), B(x)}]n

[1 + min{A(x), B(x)}]n

+ [1 min{A(x), B(x)}]n

= min

[1 + A(x)]

n [1 A(x)]n

[1 + A(x)]n

+ [1 A(x)]n ,

[1 + B(x)]n [1 B(x)]

n

[1 + B(x)]n

+ [1 B(x)]n

=AnBn(x) .

That is to say (A B)n =An Bn,

Proofs of (v) and (vi) are similar to those of (iii) and (iv).

Based on the above operations, we can define new concentration and dilation of

an AIFS as follows.

Definition 7. The concentration of an AIFS A in the universe E is denoted by

CON(A) and is defined by

CON(A) ={(x, CON(A)(x), CON(A)(x))|x E} (17)

where

CON(A)(x) = 2[A(x)]

2

[2 A(x)]2

+ [A(x)]2

, CON(A)(x) = [1 + A(x)]

2

[1 A(x)]2

[1 + A(x)]2

+ [1 A(x)]2

.

In other words, concentration of an AIFS is defined by CON(A) =A2.

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Definition 8. The dilation of an AIFSA in the universeEis denoted by DIL(A)

and is defined by

DIL(A) ={(x, DIL(A)(x), DIL(A)(x))|x E} (18)

where

DIL(A)(x) = 2[A(x)]

1/2

[2 A(x)]1/2

+ [A(x)]1/2

, DIL(A)(x) = [1 + A(x)]

1/2

[1 A(x)]1/2

[1 + A(x)]1/2

+ [1 A(x)]1/2

In other words, dilation of an AIFS is defined by DIL(A) =A1/2.

The following propositions are now straightforward.

Proposition 4. For an AIFS A

of the universeE

.

(i) CON(A) A DIL(A),

(ii) IfA(x) = 0, thenCON(A)(x) = 0,

(iii) IfA(x) = 0, thenDIL(A)(x) = 0,

(iv) CON(A) = CON(A),

(v) CON(A) = CON(A),

(vi) DIL(A) = DIL(A),

(vii) DIL(A) = DIL(A).

To represent the values of linguistic variables, some terms, such as the negation

not, the connectives and and or, the hedges very, highly and more or

less, etc., are viewed as labels of various operations over AIFSs in the universe of

discourse as follows.12

not(A) = A,

Aand B =A B={(x, min{A(x), B(x)}, max{A(x), B(x)})|x E},

Aor B =A B ={(x, max{A(x), B(x)}, min{A(x), B(x)})|x E},

very (A) = CON(A),

more or less (A) = DIL(A) .

Based on the Einstein operations, we define here very, more or less,

highly, plus, minus, etc. IfAbe an AIFS, then

CON(A) may be treated as very(A),

DIL(A)may be treated as more or less(A),

and furthermore we have

plus(A) =A1.25,

minus(A) =A0.75,

highly(A) = minus very very(A),

plus plus(A) = minus very(A),

highly(A) = plus plus very(A).

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Example 2. Let the universe be given byE={6, 7, 8, 9, 10}, an AIFS LARGEofA may be defined by LARGE = { (0.1, 0.8),(0.3, 0.5),(0.6, 0.2),(0.9, 0), (1, 0)},

then

Very LARGE = {(0.0055, 0.9756), (0.0604, 0.8), (0.3103, 0.3846), (0.802, 0), (1, 0)},

Not very LARGE = {(0.9756, 0.0055), (0.8, 0.0604), (0.3846, 0.3103), (0, 0.802), (0, 1)},

Very very LARGE = {(0, 0.9997), (0.0019, 0.9756), (0.0653, 0.6701), (0.6189, 0), (1, 0)},

More or less LARGE = {(0.3732, 0.5), (0.5916, 0.2679), (0.7913, 0.101), (0.9499, 0), (1, 0)}.

Example 3. Let A be the AIFS denoting the linguistic variable YOUNG with

the universe E= [0, 100], and furthermore

A(YOUNG) ={(x, YOUNG(x), YOUNG(x)) |x E}

where

YOUNG(x) =

1, x [0, 25],

1 +

x 25

5

21, x [25, 100],

YOUNG(x) =

0, x [0, 28],

1

1 +

x 28

5

21, x [28, 100].

then

very YOUNG ={(x, veryYOUNG(x), very YOUNG(x))|x E}

where

veryYOUNG(x) =

1, x [0, 25],

2 [(1 + (x255 )2)1]2

[2 (1 + (x255

)2)1]2+

1 +

x 25

5

212, x [25, 100]

and

veryYOUNG(x) =

0, x [0, 28],

[2 (1 + (x285 )2)1]2 [(1 + (x285 )

2)1]2

[2 (1 + (

x28

5 )2

)1

]2

+ [(1 + (

x28

5 )2

)1

]2

, x [28, 100].

and more or less YOUNG is defined by

more or less YOUNG ={(x, moreor lessYOUNG(x), more or lessYOUNG(x))|x E}

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where

more or lessYOUNG(x)

=

1, x [0, 25],

2 [(1 + (x255 )2)1]1/2

[2 (1 + (x255 )2)1]1/2 + [(1 + (x255 )

2)1]1/2, x [25, 100]

and

more or lessYOUNG(x)

=

0, x [0, 28],

[2 (1 + (x285 )2)1]1/2 [(1 + (x285 )

2)1]1/2

[2 (1 + (x285 )2)1]1/2 + [(1 + (x285 )

2)1]1/2, x [28, 100].

4. Conclusion

In this paper we have defined new concentration (CON(A)) and dilation

(DIL(A)), which will be useful in the calculus of linguistic variables under in-

tuitionistic fuzzy environment. We see that CON

(A) A DIL

(A), CON

(A)could be viewed as very(A) while DIL(A) as more or less (A). We have made

the study with suitable examples. It is worth to point out that the proposed Einstein

operations over AIFSs will be applied to aggregating intuitionistic fuzzy information

in the future.

Acknowledgements

This work was supported by the National Natural Science Foundation of China

(NSFC) (71171048), the Scientific Research and Innovation Project for College

Graduates of Jiangsu Province (CXZZ11 0185), the Scientific Research Founda-

tion of Graduate School of Southeast University (YBJJ1135) and the State Key

Laboratory of Rail Traffic Control and Safety (RCS2011K002) of Beijing Jiaotong

University.

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