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The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets
Information Sciences xxx (2014) xxx–xxx
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The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets q Xue-hai Yuan a,⇑, Hong-xing Li a, Cheng Zhang b a b
School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, PR China School of Information Engineering, Dalian University, Dalian 116622, PR China
a r t i c l e
i n f o
Article history: Received 30 June 2008 Received in revised form 6 January 2009 Accepted 6 February 2014 Available online xxxx Keywords: Intuitionistic fuzzy special set Intuitionistic fuzzy set Intuitionistic nested set Cut set Decomposition theorem Representation theorem
a b s t r a c t In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of k-cut sets of IFS and show that k-cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of k-cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction As an extension of Zadeh’s fuzzy set, Atanassov introduced the concept of intuitionistic fuzzy set (IFS) in 1986 [1]. Since then, the theories and applications on IFS such as intuitionistic fuzzy logic [2,3], intuitionistic fuzzy numbers [20], intuitionistic fuzzy algebra [4,5,25,27], intuitionistic fuzzy topology [6,9,21,23], intuitionistic fuzzy clustering [11,26], pattern recognition based on IFS [12,13], state machine theory based on IFS [28] and decision making based on IFS [14,16,17,22], etc., have been established by researchers. Recently, the concept of the intuitionistic fuzzy special set (IFSS) of a set X was introduced in [8]. An IFSS is an object A ¼ hX; A1 ; A2 i, where A1 X; A2 X and A1 \ A2 ¼ ;. Based on the concept of IFSS, intuitionistic fuzzy special points, intuitionistic fuzzy special topological spaces and intuitionistic fuzzy special r-algebras were introduced in [8,21]. These works impel us to reveal more connections between the IFS and the IFSS. It is well known that fuzzy sets presented by Zadeh [29] have the intimate connections with the classical sets, a fuzzy set can be considered as an equivalent class of a nested set [10,18,19]. By using the concept of the nested set, one can build up the decomposition theorem and the representation theorem of fuzzy sets and give the membership function of a fuzzy set [18]. The theory of the nested set has built up a theoretical approach for Zadeh’s fuzzy sets. In [15,30], Li and Zou, etc., gave the concept of ða; bÞ-cut set of an IFS and built up the decomposition theorem and the representation theorem of IFS based on ða; bÞ-cut set of IFS. However, we will point in Section 4 and Section 5 of this paper that ða; bÞ-strong cut set of IFS does not preserve the operation ‘‘Union ([)’’ and the representation theorem given in [30] do not preserve operation
q
This research was supported by the National Science Foundation of China (60774049).
⇑ Corresponding author. Tel.: +86 411 84706542; fax: +86 411 84706405. E-mail address: [email protected] (X.-h. Yuan). http://dx.doi.org/10.1016/j.ins.2014.02.044 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: X.-h. Yuan et al., The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.044
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‘‘Complement(c)’’. In order to improve the situations and build up a theoretical approach for IFS, it is needed to present a more reasonable definition of cut sets on IFS. The aim of this paper is to build up a connection between IFS and IFSS. In Section 3, we give the concept of intuitionistic nested sets (INS) and show that an IFS is an equivalent class of an INS. By using the concept of INS, we can determine the degree of membership lA ðxÞ and the degree of non-membership mA ðxÞ of an element x. In Section 4, by considering the cut set of an IFS as an IFSS, we present the concept of k-cut set of IFS and show that the k-cut set has the same properties as the cut set of Zadeh’s fuzzy set. In Section 5, by using theory of k-cut set and INS, we establish the decomposition theorem, representation theorem and extension principle of IFS. In Section 6, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup. 2. Preliminary
Definition 2.1. [1] Let X be a set and lA : X ! ½0; 1; mA : X ! ½0; 1 be two functions satisfying lA ðxÞ þ mA ðxÞ 6 1; 8x 2 X. Then A ¼ hX; lA ; mA i is called an intuitionistic fuzzy set of X. The class of all intuitionistic fuzzy sets of X is denoted as LX . For A ¼ hX; lA ; mA i, B ¼ hX; lA ; mA i, AðcÞ ¼ hX; lAðcÞ ; mAðcÞ i 2 LX . Then we have the following operations:
A B () ðlA ðxÞ 6 lB ðxÞ;
mA ðxÞ P mB ðxÞÞ; 8x 2 X;
A ¼ B () ðlA ðxÞ ¼ lB ðxÞ; mA ðxÞ ¼ mB ðxÞÞ; 8x 2 X; Ac ¼ hX; mA ; lA i; \ AðcÞ ¼ X; \ lAðcÞ \ mAðcÞ ; ¼ X; [ lAðcÞ ; \ mAðcÞ : c2C
c2C
c2C
c2C
c2C
Definition 2.2. [8] Let X be a fixed nonempty set. An intuitionistic fuzzy special set (IFSS) A is an object with the form
A ¼ hX; A1 ; A2 i; where A1 X; A2 X and A1 \ A2 ¼ ;. The set A1 is called the set of members of A, while A2 is called the set of nonmembers of A. Obviously, each subset C of X can form an IFSS having the form hX; C; C c i. Coskun gave the following definition. Definition 2.3. [8] Let A and B be two IFSSs with the form A ¼ hX; A1 ; A2 i; B ¼ hX; B1 ; B2 i respectively. Let {Ai j i 2 Ig be an arbitrary family of IFSSs on X, where Ai ¼ hX; A1i ; A2i i. Then
ðaÞ A BiffA1 B1 andA2 B2 ; ðbÞ A ¼ BiffA BandB A; D E ðdÞ ; ¼ X; ;; Xi; X ¼ hX; X; ; ; ðcÞ Ac ¼ hX; A2 ; A1 i; 1 2 1 ðfÞ \ Ai ¼ X; \ Ai ; \ A2i ; ðeÞ \ Ai ¼ X; \ Ai ; \ Ai ; i2I
i2I
i2I
i2I
i2I
i2I
ðgÞ A n B ¼ A \ Bc : Definition 2.4. [8] Let f : X ! Y be a function. (a) If B ¼ hY; B1 ; B2 i is an IFSS on Y, then the preimage of B under f, denoted by f 1 ðBÞ, is an IFSS on X defined by
f 1 ðBÞ ¼ hX; f 1 ðB1 Þ; f 1 ðB2 Þi: (b) If A ¼ hX; A1 ; A2 i is an IFSS on X, then the image of A under f, denoted by f ðAÞ, is an IFSS on Y defined by
f ðAÞ ¼ hX; f ðA1 Þ; f ðA2 Þi; c
where f ðA1 Þ ¼ ff ðxÞ j x 2 A1 g; f ðA2 Þ ¼ Y f ðX n A2 Þ ¼ ðf ðAc2 ÞÞ . Definition 2.5. [18] Let PðXÞ be the power set of X. If mapping H : ½0; 1 ! PðXÞ satisfies
k1 < k2 ) Hðk2 Þ Hðk1 Þ; then H is called a nested set over X. Definition 2.6. [7,19] Let A : X ! ½0; 1 be a fuzzy subset of X. For k 2 ½0; 1,
Ak ¼ fx j x 2 X; AðxÞ P kg;
Ak ¼ fx j x 2 X; AðxÞ > kg;
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are called k-cut set and k-strong cut set of fuzzy set A respectively. Definition 2.7. [15,30] Let A ¼ hX; lA ; mA i be an IFS, a; b 2 ½0; 1 and a þ b 6 1. Then
Aða;bÞ ¼ fx j x 2 X; lA ðxÞ P a and mA ðxÞ 6 bg; Aða;bÞ ¼ fx j x 2 X; lA ðxÞ > a and mA ðxÞ < bg; are call ða; bÞ-upper cut set and ða; bÞ-strong upper cut set of A respectively [15].
Aða;bÞ ¼ fx j x 2 X; lA ðxÞ 6 a and A
ða;bÞ
mA ðxÞ P bg; ¼ fx j x 2 X; lA ðxÞ < a and mA ðxÞ > bg;
are called ða; bÞ-lower cut set and ða; bÞ-strong lower cut set of A respectively [30]. 3. The extension from IFSS to IFS Let FPðXÞ ¼ fA j A ¼ hX; A1 ; A2 i is an IFSS over X}, we first give the definition of intuitionistic nested set. Definition 3.1. If mapping H : ½0; 1 ! FPðXÞ satisfies:
k1 < k2 ) Hðk2 Þ Hðk1 Þ; then H is called an intuitionistic nested set (INS) over X. Note 3.1. Let HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi. Then we have
ð1Þ Hl ðkÞ \ Hm ðkÞ ¼ ;; ð2Þ k1 < k2 ) Hl ðk1 Þ Hl ðk2 Þ; Hm ðk1 Þ Hm ðk2 Þ: Example 3.1. Let X ¼ fx1 ; x2 ; x3 ; x4 g and
8 hX; fx1 ; x2 ; x3 ; x4 g; ;i; > > > < hX; fx ; x g; fx gi; 2 3 4 HðkÞ ¼ > hX; fx3 g; fx4 gi; > > : hX; ;; fx1 ; x4 gi;
0 6 k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 1
Then we have that
ðaÞ Hl ðkÞ \ Hm ðkÞ ¼ ;; ðbÞ k1 < k2 ) Hl ðk1 Þ Hl ðk2 Þ; Hm ðk1 Þ Hm ðk2 Þ: Thus H is an INS over X. Let UPðXÞ ¼ fH j H is an INS over Xg. We define the operations in UPðXÞ as follows:
[ Hc :
c2C
c2C
\ Hc :
c2C
[ Hc ðkÞ ¼ [ Hc ðkÞ; i:e:; [ Hc ðkÞ ¼ [ ðHc Þl ðkÞ; [ Hc ðkÞ ¼ \ ðHc Þm ðkÞ c2C
c2C
c2C
l
c2C
m
c2C
\ Hc ðkÞ ¼ \ Hc ðkÞ; i:e:; \ Hc ðkÞ ¼ \ ðHc Þl ðkÞ; \ Hc ðkÞ ¼ [ ðHc Þm ðkÞHc : ðHc ÞðkÞ
c2C
c2C
c
c2C
c
c2C
l
c2C
m
c2C
c
¼ ðHð1 kÞÞ ; i:e:; ðH Þl ðkÞ ¼ Hm ð1 kÞ; ðH Þm ðkÞ ¼ Hl ð1 kÞ: Then we have the following conclusion. Theorem 3.1. ðUPðXÞ; [; \; cÞ is a complete De Morgan algebra, i.e., ðUPðXÞ; [; \; cÞ is a complete lattice and c
ð1Þ ðHc Þ ¼ H; c c \ Hc ¼ [ ðHc Þc ; ð2Þ [ Hc ¼ \ ðHc Þc ; c2C c2C c2C c2C ð3Þ H \ [ Hc ¼ [ ðH \ Hc Þ; H [ \ Hc ¼ \ ðH [ Hc Þ: c2C
c2C
c2C
c2C
Please cite this article in press as: X.-h. Yuan et al., The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.044
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Clearly, if we set
H1 H2 () H1 ðkÞ H2 ðkÞ; 8k 2 ½0; 1; then is a partial order and H1 H2 iff H1 [ H2 ¼ H2 iff H1 \ H2 ¼ H1 . If we define the relation \ " in UPðXÞ as
H1 H2 () \ H1 ðaÞ ¼ \ H2 ðaÞ; 8k 2 ð0; 1; a
a
i.e.,
H1 H2 () \ ðH1 Þl ðaÞ ¼ \ ðH2 Þl ðaÞ; [ ðH1 Þm ðaÞ ¼ [ ðH2 Þm ðaÞ; a
a
a
a
then is an equivalent relation over UPðXÞ. For H 2 UPðXÞ; we let ½H ¼ fH0 j H0 2 UPðXÞ; H0 Hg and IF 0 ðXÞ ¼ f½H j H 2 UPðXÞg. Let
F H : ½0; 1 ! FPðXÞ k # F H ðkÞ ¼ \ HðaÞ; F H ð0Þ ¼ X ¼ hX; X; ;i; a
F H : ½0; 1 ! FPðXÞ k # F H ðkÞ ¼ [ HðaÞ; F H ð1Þ ¼ ; ¼ hX; ;; Xi: a>k
Then we have the following conclusions. Lemma 3.1. (1) (2) (3) (4) (5)
F H ; F H 2 UPðXÞ; k1 < k2 ) F H ðk1 Þ F H ðk2 Þ; FH H FH; F H ðkÞ ¼ [a>k F H ðaÞ; 8k 2 ½0; 1Þ; F H ðkÞ ¼ \ak F H ðaÞ; 8k 2 ½0; 1Þ; F H ðkÞ ¼ \a
Proof. (1) is obvious. (2) Let k1 < k2 and a0 2 ð0; 1Þ such that k1 < a0 < k2 . Then we have that
F H ðk1 Þ ¼ [ HðaÞ Hða0 Þ \ HðaÞ ¼ F H ðk2 Þ: a>k1
a
(3) By HðaÞ HðkÞ; 8a > k, we have that F H ðkÞ ¼ [a>k HðaÞ HðkÞ. Similarly, we have that F H ðkÞ ¼ \a kðk – 1Þ, we know that [a>k F H ðaÞ F H ðkÞ. On the other hand, [a>k F H ðaÞ [a>k HðaÞ ¼ F H ðkÞ. Therefore, F H ðkÞ ¼ [a>k FðaÞ; 8k 2 ½0; 1Þ. The proofs of others are similar. h Lemma 3.2. Let H; H0 2 UPðXÞ. Then
H H0 () [ HðaÞ ¼ [ H0 ðaÞ; 8k 2 ½0; 1Þ: a>k
a>k
Proof. We only need to prove F H ðaÞ ¼ F H0 ðaÞ; 8a 2 ð0; 1. Then
that
8a 2 ð0; 1; F H ðaÞ ¼ F H0 ðaÞ () 8a 2 ½0; 1Þ; F H ðaÞ ¼ F H0 ðaÞ.
In
fact,
Let
F H ðkÞ ¼ [ F H ðaÞ ¼ [ F H0 ðaÞ ¼ F H0 ðkÞ; 8k 2 ½0; 1Þ: a>k
a>k
On the other hand, let F H ðaÞ ¼ F 0H ðaÞ; 8a 2 ð0; 1. Then
F H ðkÞ ¼ \ F H ðaÞ ¼ \ F H0 ðaÞ ¼ F H0 ðkÞ; 8k 2 ð0; 1: a
a
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Lemma 3.3. Let Hc H0c ; H H0 ðc 2 CÞ. Then
\ Hc \ H0c ; [ Hc [ H0c ; Hc H0c :
c2C
c2C
c2C
c2C
Proof. (1) Let Hc H0c ðc 2 CÞ, i.e., \a
\
\ Hc ðaÞ ¼ \ \ Hc ðaÞ ¼ \ \ Hc ðaÞ ¼ \ \ H0c ðaÞ ¼ \ \ H0c ðaÞ:
a
a
c2Ca
c2Ca
a
0
It follows that \c2C Hc \c2C Hc . (2) Let Hc H0c ðc 2 CÞ, i.e., [a>k Hc ðaÞ ¼ [a>k H0c ðaÞ; 8k 2 ½0; 1Þ. Then
[
[ Hc ðaÞ ¼ [ [ Hc ðaÞ ¼ [ [ Hc ðaÞ ¼ [ [ H0c ðaÞ ¼ [ [ H0c ðaÞ:
a>k c2C
a>kc2C
c2Ca>k
c2Ca>k
a>k c2C
0
It follows that [c2C Hc [c2C Hc . (3) Let H H0 ; i:e:; [a>k HðaÞ ¼ [a>k HðaÞ; 8k 2 ½0; 1Þ. Then c
c
c
\a1k Hð1 aÞÞc ¼ ð[1a>1k H0 ð1 aÞÞ ¼ \a
[ ½Hc ¼
c2C
[ Hc ; \ ½Hc ¼ \ Hc ; ½Hc ¼ ½Hc :
c2C
c2C
c2C
By Lemma 3.3, we know that Definition 3.2 is well-defined. Let A ¼ hX; A1 ; A2 i 2 FPðXÞ; HA ðkÞ hX; A1 ; A2 i; 8k 2 ½0; 1 and FP 0 ðXÞ ¼ f½HA j A 2 FPðXÞg. Then ðFP 0 ðXÞ; [; \; cÞ is isomorphic with ðFPðXÞ; [; \; cÞ and ðFP 0 ðXÞ IF 0 ðXÞ. Let IF ðXÞ ¼ ðIF 0 ðXÞ FP0ðXÞÞ [ FPðXÞ, then we obtain a new algebra system ðIF ðXÞ; [; \; cÞ, and ðIF ðXÞ; [; \; cÞ is isomorphic with system ðIF 0 ðXÞ; [; \; cÞ by embedding theorem of algebra [24]. Theorem 3.2. Let f : IF ðXÞ ! LX ,
A ¼ ½H # hX; lA ; mA i; where
lA ðxÞ ¼ _fk j x 2 Hl ðkÞg; mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg. Then f is a surjection.
Proof. Firstly, we show that f is well-defined. Let A ¼ ½H ¼ ½H0 . Then F H ðkÞ ¼ F H0 ðkÞ H0 ðkÞ F H0 ðkÞ ¼ F H ðkÞ; 8k 2 ½0; 1, i.e.,
ðF H Þl ðkÞ ¼ ðF H0 Þl ðkÞ ðH0 Þl ðkÞ ðF H0 Þl ðkÞ ¼ ðF H Þl ðkÞ; ðF H Þm ðkÞ ¼ ðF H0 Þm ðkÞ ðH0 Þm ðkÞ ðF H0 Þm ðkÞ ¼ ðF H Þm ðkÞ: Thus
_fk j x 2 ðF H Þl ðkÞg 6 _fk j x 2 ðH0 Þl ðkÞg 6 fk j x 2 ðF H Þl ðkÞg: Let d 2 fk j x 2 ðF H Þl ðkÞg: Then x 2 ðF H Þl ðdÞ. By Lemma 3.1(2) we have that x 2 ðF H Þl ðdÞ ðF H Þl ðkÞ; 8k < d. It follows that _fk j x 2 ðF H Þl ðkÞg P _fk j k < dg ¼ d. Thus _fk j x 2 ðF H Þl ðkÞg P _fd j x 2 ðF H Þl ðdÞg. Then
_fk j x 2 ðF H Þl ðkÞg ¼ _fk j x 2 ðH0 Þl ðkÞg ¼ _fk j x 2 ðF H Þl ðkÞg: Therefore, lA ðxÞ ¼ _fk j x 2 Hl ðkÞg ¼ _fk j x 2 ðH0 Þl ðkÞg. Similarly,
^f1 k j x R ðF H Þm ðkÞg P ^f1 k j x R ðH0 Þm ðkÞg P ^f1 k j x R ðF H Þm ðkÞg Let d 2 fk j x R ðF H Þm ðkÞg. By Lemma 3.1(2), we have that x R ðF H Þm ðdÞ ðF H Þm ðkÞ; 8k < d, and consequently x R ðF H Þm ðkÞ; 8k < d. Then ^f1 k j x R ððFÞH Þm ðkÞg 6 ^f1 k j k < dg ¼ 1 d. Thus ^f1 k j x R ðF H Þm ðkÞg 6 ^f1 d j x R ðF H Þm ðdÞg. It follows that
^f1 k j x R ðF H Þm ðkÞg ¼ ^f1 k j x R ðH0 Þm ðkÞg ¼ ^f1 k j x R ðF H Þm ðkÞg:
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Thus
mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg ¼ ^f1 k j x R ðH0 Þm ðkÞg: Therefore, f is well-defined. Secondly, we prove that lA ðxÞ þ mA ðxÞ 6 1; 8x 2 X. In fact, assume that
lA ðxÞ þ mA ðxÞ > 1, then we have that
_fk j x 2 Hl ðkÞg > 1 ^f1 k j x R Hm ðkÞg ¼ _fk j x R Hm ðkÞg: Thus there exists a k0 2 ½0; 1 such that x 2 Hl ðk0 Þ and k0 > _fk j x R Hm ðkÞg. From Hl ðk0 Þ \ Hm ðk0 Þ ¼ ;, we have x0 R Hm ðk0 Þ. It follows that k0 > _fk j x R Hm ðkÞg P k0 , which leads to a contradiction. Finally, we show that f is a surjection. Let hX; lA ; mA i 2 LX and ðHA ÞðkÞ ¼ hX; ðHA Þl ðkÞ; ðHA Þm ðkÞi, where ðHA Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg and ðHA Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg; 8k 2 ½0; 1. Then we have that HA 2 UPðXÞ and _fk j x 2 ðHA Þl ðkÞg ¼ lA ðxÞ; ^f1 k j x R ðHA Þm ðkÞg ¼ mA ðxÞ. Thus f ð½HA Þ ¼ hX; lA ; mA i. Therefore f is a surjection. h Theorem 3.3. Let A ¼ ½H 2 IF ðXÞ and hX; lA ; mA i ¼ f ð½HÞ. Then
ð1Þ ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg; ð2Þ ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg; ð3Þ ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ > kg; ð4Þ ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ P 1 kg:
Proof. (1) If x 2 ðF H Þl ðkÞ, then we have that lA ðxÞ ¼ _fa j x 2 ðF H Þl ðaÞg P k. On the other hand, if x R ðF H Þl ðkÞ ¼ \a
mA ðxÞ ¼ ^f1 a j x R ðHÞm ðaÞg P ^f1 a j a 6 a0 g ¼ 1 a0 > 1 k: On the other hand, x R ðF H Þm ðkÞ ) x R ðF H Þm ðaÞ; 8a < k ) mA ðxÞ ¼ ^f1 a j x R ðF H Þm ðaÞg 6 ^f1 a j a < kg ¼ 1 k. It follows that ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg. Proofs of (3) and (4) are similar.
h
Theorem 3.4. ðIF ðXÞ; [; \; cÞffif ðLX ; _; ^; cÞ; i.e., f is a bijection and satisfies that
\ ½Hc ¼ \ f ð½Hc Þ; c2C c2C ð2Þ f [ ½Hc ¼ [ f ð½Hc Þ;
ð1Þ f
c2C
c2C
c
ð3Þ f ð½Hc Þ ¼ ðf ð½HÞÞ : Proof. (1) Let f ð½HÞ ¼ f ð½H0 Þ ¼ hX; lA ; mA i. Then
ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg ¼ ðF H0 Þl ðkÞ; 8k 2 ½0; 1; ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg ¼ ðF H0 Þm ðkÞ; 8k 2 ½0; 1: Thus F H ðkÞ ¼ F H0 ðkÞ; 8k 2 ½0; 1 and consequently H H0 , i.e., ½H ¼ ½H0 . Therefore f is a bijection. (2) Let AðcÞ ¼ ½Hc 2 IF ðXÞ; f ðAðcÞ Þ ¼ hX; lAðcÞ ; mAðcÞ i and ½H 2 IF ðXÞ such that f ½H ¼ \c2C hX; lAðcÞ ; mAðcÞ i ¼ hX; \c2C lAðcÞ ; [c2C mAðcÞ i. Then
ðF H Þl ðkÞ ¼ x j x 2 X; ^c2C lAðcÞ ðxÞ P k ¼ \ fx j x 2 X; lAðcÞ ðxÞ P kg c2C ¼ \ ðF HðcÞ Þl ðkÞ ¼ \ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C c2C l ðF H Þm ðkÞ ¼ x j x 2 X; _c2C mAðcÞ ðxÞ > 1 k ¼ [ fx j x 2 X; mAðcÞ ðxÞ > 1 kg c2C ¼ [ ðF HðcÞ Þm ðkÞ ¼ \ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
m
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7
Thus ðF H ÞðkÞ ¼ ð\c2C F HðcÞ ÞðkÞ; 8k 2 ½0; 1. It follows that
½H ¼ ½F H ¼
\ F HðcÞ ¼ \ ½F HðcÞ ¼ \ ½HðcÞ ¼ \ AðcÞ :
c2C
c2C
c2C
c2C
and
f
\ AðcÞ
c2C
ðcÞ
(3) Let A
¼ f ð½HÞ ¼ \ hX; lAðcÞ ; mAðcÞ i ¼ \ f ðAðcÞ Þ: c2C
c2C
ðcÞ
¼ ½HðcÞ 2 IF ðXÞ; f ðA Þ ¼ hX; lAðcÞ ; mAðcÞ i and ½H 2 IF ðXÞ such that
f ð½HÞ ¼ [ f ðAðcÞ Þ ¼ c2C
X; [ lAðcÞ ; \ mAðcÞ : c2C
c2C
Then
ðF H Þl ðkÞ ¼ fx j x 2 X; _c2C lAðcÞ ðxÞ > kg ¼ [ fx j x 2 X; lAðcÞ ðxÞ > kg c2C ¼ [ ðF HðcÞ Þl ðkÞ ¼ [ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
l
ðF H Þm ðkÞ ¼ fx j x 2 X; ^c2C mAðcÞ ðxÞ P 1 kg ¼ \ fx j x 2 X; mAðcÞ ðxÞ P 1 kg c2C ¼ \ ðF HðcÞ Þm ðkÞ ¼ [ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
m
Thus ½H ¼ ½F H ¼ ½[c2C F HðcÞ ¼ [c2C ½F HðcÞ ¼ [c2C ½HðcÞ ¼ [c2C AðcÞ . Therefore, f ð[c2C AðcÞ Þ ¼ [c2C f ðAðcÞ Þ. (4) Let A ¼ ½H 2 IF ðXÞ; f ð½HÞ ¼ hX; lA ; mA i and ½H 2 IF ðXÞ, such that f ð½HÞ ¼ hX; mA ; lA i. Then
ðF H Þl ðkÞ ¼ fx j x 2 X; mA ðxÞ P kg ¼ ðF H Þm ð1 kÞ and
ðF H Þm ðkÞ ¼ fx j x 2 X; lA ðxÞ > 1 kg ¼ ðF H Þl ð1 kÞ: Thus
ðF H ÞðkÞ ¼ hX; ðF H Þm ð1 kÞ; ðF H Þl ð1 kÞi ¼ hX; ðF H Þl ð1 kÞ; ðF H Þm ð1 kÞic ¼ ðF H ð1 kÞÞc ¼ ðF H Þc ðkÞ: It follows that ½H ¼ ½F H ¼ ½ðF H Þc ¼ ½F H c ¼ ½Hc . Therefore, f ð½Hc Þ ¼ f ð½HÞ ¼ f ð½HÞc . h Example 3.2. Let X ¼ fðx; yÞ j x; y 2 ½0; 1; x2 þ 2y2 6 1g. For k 2 ½0; 1, we set
HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi;
Fig. 1.
lA and Hl ðkÞ of k-cut set on IFS ðlA ; mA Þ.
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
Fig. 2.
mA and Hm ðkÞ of k-cut set on IFS ðlA ; mA Þ.
where Hl ðkÞ ¼ fðx; yÞ j x; y 2 X; x2 þ 2y2 6 1 kg; Hm ðkÞ ¼ fðx; yÞ j x; y 2 X; x2 þ 2y2 >
pffiffiffiffiffiffiffiffiffiffiffiffi 1 k. Let A ¼ TðHÞ ¼ hX; lA ; mA i, then
lA ðx; yÞ ¼ _fk j ðx; yÞ 2 Hl ðkÞg ¼ 1 ðx2 þ 2y2 Þ; 2
mA ðx; yÞ ¼ ^f1 k j ðx; yÞ R Hm ðkÞg ¼ ðx2 þ 2y2 Þ : When k ¼ 0, 0.25, 0.5, 0.75, 1, lA and kHl ðkÞ are shown in the Fig. 1(a) and (b) respectively. mA and k Hm ðkÞ are shown in the Fig. 2(a) and (b) respectively. Note 3.2. By Theorem 3.4, we know that an IFS hX; lA ; mA i presented by Atanassov can be seen as an equivalent class ½H of an INS, which is just as a real number can be seen as an equavalent class of rational number. Therefore, we will not distanguish A ¼ ½H from f ð½HÞ ¼ hX; lA ; mA i in the following discussions. 4. The cut sets of IFSs By Theorem 3.3, we can easily give the definition of k-cut set of IFS. Definition 4.1. Let A ¼ hX; lA ; mA i can be an IFS over X; k 2 ½0; 1. Let
Alk ¼ fx j x 2 X; lA ðxÞ P kg;
Am½k ¼ fx j x 2 X; mA ðxÞ > 1 kg;
Alk ¼ fx j x 2 X; lA ðxÞ > kg;
Am½k ¼ fx j x 2 X; mA ðxÞ P 1 kg:
Then Ak ¼ hX; Alk ; Am½k i and Ak ¼ hX; Alk ; Am½k i are called k-cut set and k-strong cut set of the A, respectively. Example 4.1. Let X ¼ fx1 ; x2 ; x3 ; x4 g and A ¼ hX; lA ; mA i, where
lA ¼ ð0:6; 0:4; 0:4; 0:8Þ; mA ¼ ð0:1; 0:2; 0:6; 0:1Þ: Then
8 X; > > > < fx ; x g; 1 4 l Ak ¼ > fx4 g; > > : ;;
Am½k
8 ;; > > > < fx3 g; ¼ > fx > 2 ; x3 g; > : X;
k 6 0:4 0:4 < k 6 0:8 0:6 < k 6 0:8 0:8 < k 6 1
8 X; > > > < fx ; x g; 1 4 l Ak ¼ > fx4 g; > > : ;;
k 6 0:4 0:4 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1
Am½k
8 ;; > > > < fx3 g; ¼ > > fx2 ; x3 g; > : X;
k < 0:4 0:4 6 k < 0:8 0:6 6 k < 0:8 0:8 6 k 6 1 k < 0:4 0:4 6 k < 0:8 0:8 6 k < 0:9 0:9 6 k 6 1
Thus k-cut set and k-strong cut set of the A have the following forms respectively: Please cite this article in press as: X.-h. Yuan et al., The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.044
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8 hX; X; ;i; > > > > > > > < hX; fx1 ; x4 g; fx3 gi; Ak ¼ hX; fx4 g; fx3 gi; > > > > hX; ;; fx2 ; x3 gi; > > > : hX; ;; Xi; 8 hX; X; ;i; > > > > > > > < hX; fx1 ; x4 g; fx3 gi; Ak ¼ hX; fx4 g; fx3 gi; > > > > hX; ;; fx2 ; x3 gi; > > > : hX; ;; Xi;
k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1 k < 0:4 0:4 6 k < 0:6 0:6 6 k < 0:8 0:8 6 k < 0:9 0:9 6 k 6 1
Clearly, we have the following properties. Property 4.1. (1) Let A 2 LX ; HA ðkÞ ¼ Ak ; HA ðkÞ ¼ Ak , then
HA ; HA 2 UPðXÞ; HA HA ; (2) k1 < k2 ) Ak1 Ak2 ; (3) Let A ¼ hX; lA ; mA i; B ¼ hX; lB ; mB i. Then
ðA \ BÞk ¼ Ak \ Bk ;
ðA \ BÞk ¼ Ak \ Bk ;
ðA [ BÞk ¼ Ak [ Bk ;
ðA [ BÞk ¼ Ak [ Bk ;
X
(4) Let At ; Bt 2 L ðt 2 TÞ. Then
[ At
t2T
\ At
t2T
[ ðAt Þk ; t2T
k
¼ [ ðAt Þk ;
t2T
¼ \ ðAt Þk ; t2T
k
[ At \ At
k
t2T
\ ðAt Þk ;
t2T
k
t2T
(5) ðAc Þk ¼ ðA1k Þc ; ðAc Þk ¼ ðA1k Þc ; (6) Let kt 2 ½0; 1ðt 2 TÞ and a ¼ _t2T kt ; b ¼ ^t2T kt . Then
\ Akt ¼ Aa ; [ Akt ¼ Ab ;
t2T
t2T
In general, Ak ¼ [a>k Aa ¼ [a>k Aa ; Ak ¼ \ Aa ¼ \ Aa ; a
Proof. Firstly, we show A ¼ [t2T At ; B ¼ \ At , then
that
ð[t2T At Þk [t2T ðAt Þk ,
i.e.,ð[t2T At Þlk [t2T ðAt Þlk ; ð[t2T At Þm½k \ ðAt Þm½k . t2T
In
fact,
let
t2T
x 2 [ ðAt Þlk ) 9t0 ; x 2 ðAt0 Þlk ) 9t0 ; t2T
lAt0 ðxÞ P k ) lA ðxÞ ¼ _t2T lAt ðxÞ P k ) x 2 Alk ;
x 2 Am½k ) ^t2T mAt ðxÞ ¼ mA ðxÞ > 1 k ) 8t 2 T; l
l
m
mAt ðxÞ > 1 k ) 8t 2 T; x 2 ðAt Þm½k ) x 2 \ ðAt Þm½k : t2T
m
Then Ak [t2T ðAt Þk ; A½k \t2T ðAt Þ½k . Thus ð[t2T At Þk [t2T ðAt Þk . l Secondly, we show that Aa ¼ \t2T Akt , i.e., Ala ¼ \t2T Akt ; Am½a ¼ [t2T Am½kt . In fact, l
l
x 2 \ Akt () 8t 2 T; x 2 Akt () 8t 2 T; t2T
x 2 [ Am½kt () 9t 0 ; t2T
lA ðxÞ P kt () lA ðxÞ P _kt ¼ a () x 2 Ala ;
mA ðxÞ > 1 kt0 () mA ðxÞ > ^t2T ð1 kt Þ ¼ 1 _t2T kt ¼ 1 a () x 2 Am½a :
l Then Ala ¼ \t2T Akt ; Am½a ¼ [ Am½kt . Thus Aa ¼ \t2T Akt . t2T
l
m
l
Finally, we show that ðAc Þk ¼ ðA1k Þc ; i:e:; ðAc Þk ¼ Am½1k ; ðAc Þ½k ¼ A1k . In fact,
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l
x 2 ðAc Þk () mA ðxÞ ¼ lAc ðxÞ P k ¼ 1 ð1 kÞ () x 2 Am½1k ; m
l
x 2 ðAc Þ½k () lA ðxÞ ¼ mAc ðxÞ > 1 k () x 2 A1k : l
m
l
Then ðAc Þk ¼ Am½1k ; ðAc Þ½k ¼ A1k . Thus ðAc Þk ¼ ðA1k Þc . Others are obvious. h Example 4.2. Let X ¼ fa; bg; AðaÞ ¼ ð0:4; 0:4Þ; AðbÞ ¼ ð0:4; 0:3Þ; BðaÞ ¼ ð0:4; 0:3Þ; BðbÞ ¼ ð0:5; 0:4Þ; CðaÞ ¼ ð0:5; 0:4Þ; CðbÞ ¼ ð0:4; 0:3Þ, ða1 ; b1 Þ ¼ ð0:4; 0:4Þ; ða2 ; b2 Þ ¼ ð0:3; 0:3Þ, ðk1 ; k2 Þ ¼ ð0:4; 0:4Þ. Then Ac ðaÞ ¼ ð0:4; 0:4Þ, Ac ðbÞ ¼ ð0:3; 0:4Þ; ðB [ CÞðaÞ ¼ ðB [ CÞðbÞ ¼ ð0:5; 0:3Þ, ða; bÞ ¼ ða1 ; b1 Þ ^ ða2 ; b2 Þ ¼ ð0:3; 0:4Þ; ðk1 ; k2 Þc ¼ ð0:4; 0:4Þ. By Definition 2.7, we have that ðAc Þðk1 ;k2 Þ ¼ fag; Aðk1 ;k2 Þc ¼ Að0:4;0:4Þ ¼ ;, ðB [ CÞðk1 ;k2 Þ ¼ fa; bg; Bðk1 ;k2 Þ ¼ C ðk1 ;k2 Þ ¼ ;, Aða1 ;b1 Þ ¼ Aða2 ;b2 Þ ¼ ;; Aða;bÞ ¼ fbg. Then
ðAc Þðk1 ;k2 Þ – ðAðk1 ;k2 Þc Þc ; ðB [ CÞðk1 ;k2 Þ – Bðk1 ;k2 Þ [ C ðk1 ;k2 Þ ; Aða1 ;b1 Þ [ Aða2 ;b2 Þ – Aða;bÞ : Therefore, ða; bÞ-upper cut sets of IFS presented in [15] do not satisfy the Property 4.1(3, 4, 5, 6). Similarly, we can show that ða; bÞ-lower cut sets of IFS presented in [30] do not also satisfy the Property 4.1(3, 4, 5, 6). Therefore, k-cut sets presented by us have more properties than ða; bÞ-cut sets in [15,30]. Note 4.1. By comparison with the properties of cut sets on Zadeh fuzzy sets in [18], we have known that cut sets of IFSs have the same properties as cut sets of Zadeh’s fuzzy sets.
5. Decomposition theorem, representation theorem and extension principle
Definition 5.1. Let A ¼ hX; A1 ; A2 i 2 FPðXÞ; k 2 ½0; 1. Then
ðkA1 ÞðxÞ ¼
k;
x 2 A1
0; x R A1
ðk A2 Þ ¼
1 k; x R A2 1;
x 2 A2
kA ¼ hX; kA1 ; k A2 i:
Property 5.1. A B ) kA kB; i:e:; hX; kA1 ; k A2 i hX; kB1 ; k B2 i Clearly, we have the following theorem from the discussion in Section 3. Theorem 5.1 (Decomposition Theorem). Let A ¼ hX; lA ; mA i be an IFS and Ak and Ak are k-cut set and k-strong cut set of A respectively. Let H 2 FPðXÞ such that Ak HðkÞ Ak . Then
ð1Þ A ¼ [ kAk ; k2½0;1
ð2Þ A ¼ [ kAk ; k2½0;1
ð3Þ A ¼ [ kHðkÞ and k2½0;1
ðaÞ k1 < k2 ) Hðk1 Þ Hðk2 Þ; ðbÞ Ak ¼ \ HðaÞ; and a
ðcÞ Ak ¼ [ HðaÞ: a>k
l
Proof. Let B ¼ [k2½0;1 kAk ¼ hX; [k2½0;1 kAk ; \k2½0;1 k Am½k i. Then
lB ðxÞ ¼ mB ðxÞ ¼
l l [ kAk ðxÞ ¼ _fk j x 2 Ak g ¼ _fk j lA ðxÞ P kg ¼ lA ðxÞ;
k2½0;1
n o l l \ k A½k ðxÞ ¼ ^ 1 k j x R A½k ¼ ^f1 k j mA ðxÞ 6 1 kg ¼ mA ðxÞ:
k2½0;1
Then A ¼ B ¼ [k2½0;1 kAk . (2) And (3) are obvious. h
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
Fig. 3.
lA and Hl ðkÞ in Example 4.1.
Fig. 4.
mA and Hm ðkÞ in Example 4.1.
Note 5.1. By Theorem 5.1(3), we have know that A ¼ [k2½0;1 kHðkÞ means lA ðxÞ ¼ _fk j x 2 Hl ðkÞg and x R Hm ðkÞg. Therefore, decomposition theorem has built the connections between IFS and IFSS.
11
mA ðxÞ ¼ ^f1 k j
Example 5.1. In Example 4.1, we have that
8 ðk; k; k; kÞ; > > > < ðk; 0; 0; kÞ; kAlk ¼ > ð0; 0; 0; kÞ; > > : ð0; 0; 0; 0Þ;
m
k A½k
k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 1
8 ð1 k; 1 k; 1 k; 1 kÞ; > > > < ð1 k; 1 k; 1; 1 kÞ; ¼ > ð1 k; 1; 1; 1 kÞ; > > : ð1; 1; 1; 1Þ;
k 6 0:4 0:4 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1
l
Then [k2½0;1 kAk ¼ ð_k60:4 ðk; k; k; kÞÞ _ ð_0:4
\k2½0;1 k Amk ¼ ð^k60:4 ð1 k;1 k;1 k;1 kÞÞ ^ ð^0:4
T : UPðXÞ ! LX ; H # TðHÞ ¼
X; [ kHl ðkÞ; \ k Hm ðkÞ k2½0;1
k2½0;1
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Then T is a homomorphism from ðUPðXÞ; [; \; cÞ onto ðLX ; [; \; cÞ, i.e., f is a surjection and
ð1Þ T
[ Hc
c2C
¼ [ TðHc Þ; T
c2C
\ Hc
c2C
¼ \c2C TðHc Þ; TðHc Þ ¼ ðTðHÞÞc ;
ð2Þ ðTðHÞÞk HðkÞ ðTðHÞÞk ; ð3Þ ðTðHÞÞk ¼ \ HðaÞ; ðTðHÞÞk ¼ [ HðaÞ: a
a>k
Note 5.2. Representation theorem has shown that each INS can construct an IFS and each IFS can be constructed by an INS. Example 5.2. Let X ¼ ½0; 2. For k 2 ½0; 1, we set HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi, h pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i Hm ðkÞ ¼ 0; 1 1 k [ 1 þ 1 k; 2 . Then H is an INS over X. Thus
where
Hl ðkÞ ¼
hpffiffiffi pffiffiffii k; 2 k
and
(a) For x 2 ½0; 1, we have that
n
o
pffiffiffi
lA ðxÞ ¼ _fk j x 2 Hl ðkÞg ¼ _ k j k 6 x 6 1 ¼ _fk j k 6 x2 g ¼ x2 : n
pffiffiffiffiffiffiffiffiffiffiffiffi
mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg ¼ ^ 1 k j 1 1 k 6 x 6 1
o
n pffiffiffiffiffiffiffiffiffiffiffiffio ¼^ 1kj061x6 1k
¼ ^f1 k j ð1 xÞ2 6 1 kg ¼ ð1 xÞ2 : (b) For x 2 ð1; 2, we have that
pffiffiffiffiffi
lA ðxÞ ¼ _fk j 1 < x 6 2 kg ¼ _fk j k 6 ð2 xÞ2 g ¼ ð2 xÞ2 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi mA ðxÞ ¼ ^f1 k j 1 < x 6 1 þ 1 kg ¼ ^f1 k j 0 < x 1 6
pffiffiffiffiffiffiffiffiffiffiffiffi 1 kg
¼ ^f1 k j ðx 1Þ2 6 1 kg ¼ ðx 1Þ2 : Therefore,
2 lA ðxÞ ¼ x ;
ð2 xÞ2 ;
06x61 1
mA ðxÞ ¼ ðx 1Þ2 .
When k ¼ 0, 0.25, 0.5, 0.75, 1, lA and kHl ðkÞ are represented in the Fig. 3(a) and (b), respectively. represented in the Fig. 4(a) and (b), respectively.
mA and k Hm ðkÞ are
Example 5.3. Let X ¼ fa; bg; AðaÞ ¼ ð0:4; 0:4Þ; AðbÞ ¼ ð0:4; 0:3Þ; a; b 2 ½0; 1 and a þ b 6 1, we set
Hða; bÞ ¼ A
ða;bÞ
8 > < fa; bg; a P 0:4; b 6 0:3; ¼ fag; a P 0:4; 0:3 < b 6 0:4; > : ;; else:
By the operation defined in [30], we have that
8 > < ;; b P 0:4; a 6 0:3; Hc ða; bÞ ¼ ðHðb; aÞÞc ¼ fbg; b P 0:4; 0:3 < a 6 0:4; > : fa; bg; else: By the representation theorem in [30], we have that
TðHÞðxÞ ¼ ^fða; bÞ j a; b 2 ½0; 1; a þ b 6 1; x 2 Hða; bÞg: Then it follows that TðHÞðaÞ ¼ AðaÞ ¼ ð0:4; 0:4Þ, ðTðHc ÞÞðaÞ ¼ ð0; 0:6Þ – ðTðHÞÞc ðaÞ. Thus the representation theorem in [30] does not preserve the operation c. Therefore, the representation theorem presented by us has better properties than that in [30]. By using Theorem 5.2 (representation theorem) and Definition 2.4, we can easily establish extension principle of IFS. Theorem 5.3 (Extension Principle). Let f : X ! Y be a mapping. Let
ðaÞ f : LX ! LY A ¼ hX; lA ; mA i # f ðAÞ ¼ [ kf ðAk Þ k2½0;1 l ¼ Y; [ kf ðAk Þ; \ k f ðAm½k Þ : k2½0;1
k2½0;1
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ðbÞ f 1 : LY ! LX B ¼ hY; lB ; mB i # f 1 ðBÞ ¼ [ kf 1 ðBk Þ k2½0;1 l 1 ¼ X; [ kf ðBk Þ; \ k f 1 ðBm½k Þ : k2½0;1
k2½0;1
Then for any A1 ; A2 ; At ðt 2 TÞ; A 2 LX and B1 ; B2 ; Bt ðt 2 TÞ; B 2 LY , we have that
lf ðAÞ ðyÞ ¼ _f ðxÞ¼y lA ðxÞ; mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ; ð2Þ lf 1 ðBÞ ðxÞ ¼ lB ðf ðxÞÞ; mf 1 ðBÞ ðxÞ ¼ mB ðf ðxÞÞ;
ð1Þ
ð3Þ A1 A2 ) f ðA1 Þ f ðA2 Þ; ð4Þ B1 B2 ) f 1 ðB1 Þ f 1 ðB2 Þ; ð5Þ A f 1 ðf ðAÞÞ; f ðf 1 ðBÞÞ B; c ð6Þ f 1 [ Bt ¼ [ f 1 ðBt Þ; f 1 \ Bt ¼ \ f 1 ðBt Þ; f 1 ðBc Þ ¼ ðf 1 ðBÞÞ ; t2T t2T t2T t2T ð7Þ f [ At ¼ [ f ðAt Þ; f \ At \ f ðAt Þ; f ðAÞc f ðAc Þ; t2T
t2T
t2T
t2T
ð8Þ ðf ðAÞÞk ¼ f ðAk Þ; ðf ðAÞÞk f ðAk Þ ð9Þ ðf 1 ðBÞÞk ¼ f 1 ðBk Þ; ðf 1 ðBÞÞk ¼ f 1 ðBk Þ:
mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ. In fact, let f 1 ðyÞ ¼ fx j x 2 X; f ðxÞ ¼ yg. Then
Proof. Firstly, we show that
n
n c o n c o ¼ ^ 1 k j y 2 f Am½k ¼ 1 k j 9x 2 Am½k ; f ðxÞ ¼ y n o ¼ ^ 1 k j 9x R Am½k ; f ðxÞ ¼ y ¼ ^f1 k j 9x 2 X; f ðxÞ ¼ y; mA ðxÞ 6 1 kg:
mf ðAÞ ðyÞ ¼ ^ 1 k j y R f Am½k
o
Thus, we can easily show that mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ. Secondly, we show that ðf ðAÞÞk ¼ f ðAk Þ; ðf ðAÞÞk f ðAk Þ. In fact,
ðf ðAÞÞk ¼ [ f ðAa Þ ¼ f a>k
ðf ðAÞÞk ¼ \ f ðAa Þ f
¼ f ðAk Þ;
[ Aa
a>k
a
\ Aa
a
¼ f ðAk Þ:
Others are obvious. h l
Example 5.4. Let X ¼ ½0; 2; Y ¼ ½0; 4 and f : X ! Y; x # x2 . In the Example 5.2, k-cut set of the A is Ak ¼ hX; Ak ; Am½k i, where hpffiffiffi h c c pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffii pffiffiffi2 l l Ak ¼ and f Am½k ¼ f Am½k k; 2 k and Am½k ¼ 0; 1 1 k [ 1 þ 1 k; 2 . Then f ðAk Þ ¼ k; 2 k pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffiffiffiffiffiffi2 c pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1 1k ; 1þ 1k ¼ 0; 1 1 k [ 1 þ 1 k ; 4 . Let B ¼ f ðAÞ ¼ hY; lB ; mB i, then lB ðyÞ ¼ n o l _ k j y 2 f Ak ; mB ðyÞ ¼ ^ 1 k j y R f Am½k . Thus pffiffiffiffiffiffiffiffiffiffiffiffi2 (a) When y 2 ½0; 1, we have that l ðyÞ ¼ _fk j k 6 yg ¼ y and m ðyÞ ¼ ^ 1 k j 1 6 y 1 k B B n pffiffiffi 2 o pffiffiffi 2 ¼ 1 y . ¼^ 1kj1kP 1 y n pffiffiffi2 pffiffiffi 2 o pffiffiffi 2 (b) When y 2 ð1; 4, we have that lB ðyÞ ¼ _ k j y 6 2 k ¼_ kjk6 2 y and mB ðyÞ ¼ ¼ 2 y n pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffi
2 o pffiffiffi
2 y1 y1 . ¼^ 1kj1kP ^ 1kj1
y; lB ðyÞ ¼
2
pffiffiffi 2 y ;
06y61 ; 1
mB ðyÞ ¼
pffiffiffi
2 y1 .
6. An application to intuitionistic fuzzy algebra In this section, we will present another definition of intuitionistic fuzzy subgroup by using the representation theorem (Theorem 5.2). Let G be a group and
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
SPðGÞ ¼ fhG; A1 ; A2 i j A1 \ A2 ¼ ;; A1
and ðA2 Þc are subgroup of Gg:
For our convenience, the empty set ; is also considered as a subgroup of G. Definition 6.1. If mapping H : ½0; 1 ! SPðGÞ k # HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi is an INS, then H is called a GINS over G. Definition 6.2. Let A ¼ hG; lA ; mA i is an IFS of G. If there is a GINS H over G such that A ¼ [k2½0;1 kHðkÞ ¼ hG; [k2½0;1 kHl ðkÞ; \k2½0;1 k Hm ðkÞi, then A is called an intuitionistic fuzzy subgroup of G. Theorem 6.1. An IFS A of G is an intuitionistic fuzzy subgroup of G if and only if for any x; y 2 G
lA ðxyÞ P lA ðxÞ ^ lA ðyÞ; lA ðx1 Þ P lA ðxÞ ðiiÞ mA ðxyÞ 6 mA ðxÞ _ mA ðyÞ; mA ðx1 Þ 6 mA ðxÞ
ðiÞ
ð6:1Þ ð6:2Þ
Proof. Let A ¼ [k2½0;1 kHðkÞ ¼ hG; [k2½0;1 kHl ðkÞ; \k2½0;1 k Hm ðkÞi is an intuitionistic fuzzy subgroup of G. We only need to show that mA ðxyÞ 6 mA ðxÞ _ mA ðyÞ. Since ðHm ðkÞÞc is a subgroup of G, then
f1 k j xy 2 ðHm ðkÞÞc g f1 k j x 2 ðHm ðkÞÞc g \ f1 k j y 2 ðHm ðkÞÞc g;
8x; y 2 G:
Thus
mA ðxyÞ ¼ ^f1 k j k 2 ½0; 1; xy R Hm ðkÞg ¼ ^f1 k j k 2 ½0; 1; xy 2 ðHm ðkÞÞc g 6 ^f1 k j k 2 ½0; 1; x 2 ðHm ðkÞÞc ; y 2 ðHm ðkÞÞc g ¼ ^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g: We need to show that
^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g 6 ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ: In fact, assume that there exists a a 2 ½0; 1 such that
^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g > a > ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ ð Þ Then Hm ð1 aÞðxÞ _ Hm ð1 aÞðyÞ ¼ 1. Thus x 2 Hm ð1 aÞ or y 2 Hm ð1 aÞ. When x 2 Hm ð1 aÞ, we have that x 2 Hm ðkÞ for any k P 1 a and x R Hm ðkÞ implies k < 1 a. Then. ^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1g ¼ ^f1 k j k 2 ½0; 1; x R Hm ðkÞg P ^f1 k j k < 1 ag ¼ a. Similarly, when y 2 Hm ð1 aÞ, we have that ^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g P a. Then. ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ P a. This contradicts with the formula (⁄). Hence
mA ðxyÞ 6 ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ ¼ ð^f1 k j k 2 ½0; 1; x R Hm ðkÞgÞ _ ð^f1 k j k 2 ½0; 1; y R Hm ðkÞgÞ ¼ mA ðxÞ _ mA ðyÞ: On the other hand, let A ¼ hG; lA ; mA i satisfy the conditions (i) and (ii). Let Hl ðkÞ ¼ fx j x 2 G; lA ðxÞ P kg, Hm ðkÞ ¼ fx j x 2 G; mA ðxÞ > 1 kg and HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi. Then H : ½0; 1 ! SPðGÞk # HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi is a GINS of G and
A ¼ [ kHðkÞ ¼ k2½0;1
G; [ kHl ðkÞ; \ k Hm ðkÞ : k2½0;1
k2½0;1
Therefore, A is an intuitionistic fuzzy subgroup of G. h 2
Example 6.1. Let G ¼ fe; a; b; abg be a group, where e is an identity and ab ¼ ba; a2 ¼ b ¼ e. Let HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi, where
8 0 6 k < 0:4 > < G; Hl ðkÞ ¼ fe; ag; 0:4 6 k < 0:8 > : feg; 0:8 < k 6 1 Then ðHm ðkÞÞc ¼
G; fe; ag;
Hm ðkÞ ¼
;;
0 6 k < 0:4
fb; abg; 0:4 6 k 6 1
0 6 k < 0:4 . 0:4 6 k 6 1
Thus Hl ðkÞandðHm ðkÞÞc are subgroup of G and consequently H is a GINS over G. Let A ¼ [k2½0;1 kHðkÞ ¼ hG; lA ; mA i, then we have that lA ¼ ð1; 0:8; 0:4; 0:4Þ and mA ¼ ð0; 0; 0:6; 0:6Þ. Clearly, lA and mA satisfy the formulae 6.1 and 6.2.
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
15
7. Conclusion In this paper, a theoretical base of IFS based on the concept of IFSS was established. Firstly, we showed that an IFS was an equivalent class of an INS. Thus an IFS can be determined by an INS. We also presented the concept of k-cut set of IFS and proved that the k-cut set of IFS had the same properties as cut set of Zadeh’s fuzzy set. Secondly, we built up the decomposition theorem, representation theorem and extension principle of IFS. We have shown the superiority of k-cut set and representation theorem in this paper. If the theory of Zadeh’s fuzzy sets was based on the classical sets and nested sets, then it seemed to obtain a conclusion that the theory of IFS was based on IFSS and INS. Therefore, our work presented a useful approach to the theory of IFS. In future research works on IFS, one should consider to determine the membership degree lA ðxÞ and the non-membership degree mA ðxÞ of an IFS. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20 (1986) 87–96. K. Atanassov, Remark on the intuitionistic fuzzy logics, Fuzzy Sets Syst. 95 (1998) 127–129. K. Atanassov, G. Gargov, Elements of intuitionistic fuzzy logic, Part I, Fuzzy Sets Syst. 95 (1998) 39–52. R. Biswas, Intuitionistic fuzzy subgroups, Math. Forum 10 (1989) 37–46. Bijan Davvaz, Wieslaw A. Dudek, Young Bue Jun, Intuitionistic fuzzy Hm -submodules, Inf. Sci. 176 (2006) 285–300. Dogan Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets Syst. 88 (1997) 81–89. D. Dubois, H. Prade, Fuzzy Sets and Systems-Theory and Applications, Academic Press, New york, 1980. Erdal Coskun, Systems on intuitionistic fuzzy special sets and intuitionistic fuzzy special measures, Inf. Sci. 128 (2000) 105–118. Francisco Gallego Lupianez, Nets and filters in intuitionistic fuzzy topological spaces, Inf. Sci. 176 (2006) 2396–2404. J.A. Herencia, Graded sets and points: a stratified approach to fuzzy sets and points, Fuzzy Sets Syst. 77 (1996) 191–202. W.L. Hung, J.S. Lee, W.L. Fuh, Fuzzy clustering based on intuitionistic fuzzy relations, Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 12 (4) (2004) 513–529. W.L. Hung, M. Yang, On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition, Inf. Sci. 178 (2008) 1641–1650. Ioannis K. Vlachos, George D. Sergiadis, Intuitionistic fuzzy information–application to pattern recognition, Pattern Recogn. Lett. 28 (2007) 197–206. Lin Lin, Xue-hai Yuan, Zun-qun Xia, Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets, J. Comput. Syst. Sci. 73 (2007) 84– 88. Li M, Cut sets of intuitionistic fuzzy sets, J. Liaon. Nor. Univ. 30 (2) (2007) 152–154. D.F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, J. Comput. Syst. Sci. 70 (2005) 73–85. H.W. Liu, G.J. Wang, Multicriteria decision-making methods based on intuitionistic fuzzy sets, Eur. J. Oper. Res. 179 (2007) 220–233. C.Z. Luo, Fuzzy sets and nested sets, Fuzzy Math. 4 (1983) 113–126. C.Z. Luo, Introduction to Fuzzy Sets, Beiing Normal University Press, 1989. H.B. Mitchell, Ranking intuitionistic fuzzy numbers, Int. J. Uncertain. Fuzz. Knowl.-Based Syst. 12 (3) (2004) 377–386. S. Özcag, D. Coker, A note on connectedness in intuitionistic fuzzy special topological spaces, Int. J. Math. Math. Sci. 23 (1) (1998) 45–54. A. Pankowska, M. Wygralak, General IF-sets with triangular norms and their applications to group decision making, Inf. Sci. 176 (2006) 2713–2754. Reza Saadati, Jin Han Park, On the intuitionistic fuzzy topological spaces, Chos, Solitons Fract. 27 (2006) 331–344. Thomas W. Hungerford, Algebra, Springer-Verlag, 1974. Wieslaw A Dudek, Bijan Davvaz, Young Bae Jun, On intuitiontistic fuzzy sub-hyperquasigroups of hyperquasigroups, Inf. Sci. 170 (2005) 251–262. Z.S. Xu, J. Chen, J.J. Wu, Clustering algorithm for intuitionistic fuzzy sets, Inf. Sci. 178 (2008) 3775–3790. Young Bae Jun, Intuitionistic nil radicals of intuitionistic fuzzy ideals and Euclidean intuitionistic fuzzy ideals in rings, Inf. Sci. 177 (2007) 4662–4677. Young Bae Jun, Quotient struvtures of intuitionistic fuzzy finite state machines, Inf. Sci. 177 (2007) 4977–4986. L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–353. Zou Kai-qi, Wang Zhi-ping, Hu Ming, New decomposition theorem and representation of the intuitionistic fuzzy sets, J. Dalian Univ. 29 (3) (2008) 1–4.
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets q Xue-hai Yuan a,⇑, Hong-xing Li a, Cheng Zhang b a b
School of Electronic and Information Engineering, Dalian University of Technology, Dalian 116024, PR China School of Information Engineering, Dalian University, Dalian 116622, PR China
a r t i c l e
i n f o
Article history: Received 30 June 2008 Received in revised form 6 January 2009 Accepted 6 February 2014 Available online xxxx Keywords: Intuitionistic fuzzy special set Intuitionistic fuzzy set Intuitionistic nested set Cut set Decomposition theorem Representation theorem
a b s t r a c t In this paper, we build up a connection between intuitionistic fuzzy special set (IFSS) and intuitionistic fuzzy set (IFS). Firstly, by using the concept of IFSS, we present the concept of intuitionistic nested set (INS) and show that an IFS can be determined by an INS. Secondly, we introduce the concept of k-cut sets of IFS and show that k-cut sets of IFS have the same properties as the cut sets of Zadeh fuzzy set. Thirdly, by using the concepts of k-cut set of IFS and INS, we build up the decomposition theorem, representation theorem and extension principle of IFS. Finally, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction As an extension of Zadeh’s fuzzy set, Atanassov introduced the concept of intuitionistic fuzzy set (IFS) in 1986 [1]. Since then, the theories and applications on IFS such as intuitionistic fuzzy logic [2,3], intuitionistic fuzzy numbers [20], intuitionistic fuzzy algebra [4,5,25,27], intuitionistic fuzzy topology [6,9,21,23], intuitionistic fuzzy clustering [11,26], pattern recognition based on IFS [12,13], state machine theory based on IFS [28] and decision making based on IFS [14,16,17,22], etc., have been established by researchers. Recently, the concept of the intuitionistic fuzzy special set (IFSS) of a set X was introduced in [8]. An IFSS is an object A ¼ hX; A1 ; A2 i, where A1 X; A2 X and A1 \ A2 ¼ ;. Based on the concept of IFSS, intuitionistic fuzzy special points, intuitionistic fuzzy special topological spaces and intuitionistic fuzzy special r-algebras were introduced in [8,21]. These works impel us to reveal more connections between the IFS and the IFSS. It is well known that fuzzy sets presented by Zadeh [29] have the intimate connections with the classical sets, a fuzzy set can be considered as an equivalent class of a nested set [10,18,19]. By using the concept of the nested set, one can build up the decomposition theorem and the representation theorem of fuzzy sets and give the membership function of a fuzzy set [18]. The theory of the nested set has built up a theoretical approach for Zadeh’s fuzzy sets. In [15,30], Li and Zou, etc., gave the concept of ða; bÞ-cut set of an IFS and built up the decomposition theorem and the representation theorem of IFS based on ða; bÞ-cut set of IFS. However, we will point in Section 4 and Section 5 of this paper that ða; bÞ-strong cut set of IFS does not preserve the operation ‘‘Union ([)’’ and the representation theorem given in [30] do not preserve operation
q
This research was supported by the National Science Foundation of China (60774049).
⇑ Corresponding author. Tel.: +86 411 84706542; fax: +86 411 84706405. E-mail address: [email protected] (X.-h. Yuan). http://dx.doi.org/10.1016/j.ins.2014.02.044 0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
‘‘Complement(c)’’. In order to improve the situations and build up a theoretical approach for IFS, it is needed to present a more reasonable definition of cut sets on IFS. The aim of this paper is to build up a connection between IFS and IFSS. In Section 3, we give the concept of intuitionistic nested sets (INS) and show that an IFS is an equivalent class of an INS. By using the concept of INS, we can determine the degree of membership lA ðxÞ and the degree of non-membership mA ðxÞ of an element x. In Section 4, by considering the cut set of an IFS as an IFSS, we present the concept of k-cut set of IFS and show that the k-cut set has the same properties as the cut set of Zadeh’s fuzzy set. In Section 5, by using theory of k-cut set and INS, we establish the decomposition theorem, representation theorem and extension principle of IFS. In Section 6, we apply our theory to intuitionistic fuzzy algebra and obtain the concept of intuitionistic fuzzy subgroup. 2. Preliminary
Definition 2.1. [1] Let X be a set and lA : X ! ½0; 1; mA : X ! ½0; 1 be two functions satisfying lA ðxÞ þ mA ðxÞ 6 1; 8x 2 X. Then A ¼ hX; lA ; mA i is called an intuitionistic fuzzy set of X. The class of all intuitionistic fuzzy sets of X is denoted as LX . For A ¼ hX; lA ; mA i, B ¼ hX; lA ; mA i, AðcÞ ¼ hX; lAðcÞ ; mAðcÞ i 2 LX . Then we have the following operations:
A B () ðlA ðxÞ 6 lB ðxÞ;
mA ðxÞ P mB ðxÞÞ; 8x 2 X;
A ¼ B () ðlA ðxÞ ¼ lB ðxÞ; mA ðxÞ ¼ mB ðxÞÞ; 8x 2 X; Ac ¼ hX; mA ; lA i; \ AðcÞ ¼ X; \ lAðcÞ \ mAðcÞ ; ¼ X; [ lAðcÞ ; \ mAðcÞ : c2C
c2C
c2C
c2C
c2C
Definition 2.2. [8] Let X be a fixed nonempty set. An intuitionistic fuzzy special set (IFSS) A is an object with the form
A ¼ hX; A1 ; A2 i; where A1 X; A2 X and A1 \ A2 ¼ ;. The set A1 is called the set of members of A, while A2 is called the set of nonmembers of A. Obviously, each subset C of X can form an IFSS having the form hX; C; C c i. Coskun gave the following definition. Definition 2.3. [8] Let A and B be two IFSSs with the form A ¼ hX; A1 ; A2 i; B ¼ hX; B1 ; B2 i respectively. Let {Ai j i 2 Ig be an arbitrary family of IFSSs on X, where Ai ¼ hX; A1i ; A2i i. Then
ðaÞ A BiffA1 B1 andA2 B2 ; ðbÞ A ¼ BiffA BandB A; D E ðdÞ ; ¼ X; ;; Xi; X ¼ hX; X; ; ; ðcÞ Ac ¼ hX; A2 ; A1 i; 1 2 1 ðfÞ \ Ai ¼ X; \ Ai ; \ A2i ; ðeÞ \ Ai ¼ X; \ Ai ; \ Ai ; i2I
i2I
i2I
i2I
i2I
i2I
ðgÞ A n B ¼ A \ Bc : Definition 2.4. [8] Let f : X ! Y be a function. (a) If B ¼ hY; B1 ; B2 i is an IFSS on Y, then the preimage of B under f, denoted by f 1 ðBÞ, is an IFSS on X defined by
f 1 ðBÞ ¼ hX; f 1 ðB1 Þ; f 1 ðB2 Þi: (b) If A ¼ hX; A1 ; A2 i is an IFSS on X, then the image of A under f, denoted by f ðAÞ, is an IFSS on Y defined by
f ðAÞ ¼ hX; f ðA1 Þ; f ðA2 Þi; c
where f ðA1 Þ ¼ ff ðxÞ j x 2 A1 g; f ðA2 Þ ¼ Y f ðX n A2 Þ ¼ ðf ðAc2 ÞÞ . Definition 2.5. [18] Let PðXÞ be the power set of X. If mapping H : ½0; 1 ! PðXÞ satisfies
k1 < k2 ) Hðk2 Þ Hðk1 Þ; then H is called a nested set over X. Definition 2.6. [7,19] Let A : X ! ½0; 1 be a fuzzy subset of X. For k 2 ½0; 1,
Ak ¼ fx j x 2 X; AðxÞ P kg;
Ak ¼ fx j x 2 X; AðxÞ > kg;
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
are called k-cut set and k-strong cut set of fuzzy set A respectively. Definition 2.7. [15,30] Let A ¼ hX; lA ; mA i be an IFS, a; b 2 ½0; 1 and a þ b 6 1. Then
Aða;bÞ ¼ fx j x 2 X; lA ðxÞ P a and mA ðxÞ 6 bg; Aða;bÞ ¼ fx j x 2 X; lA ðxÞ > a and mA ðxÞ < bg; are call ða; bÞ-upper cut set and ða; bÞ-strong upper cut set of A respectively [15].
Aða;bÞ ¼ fx j x 2 X; lA ðxÞ 6 a and A
ða;bÞ
mA ðxÞ P bg; ¼ fx j x 2 X; lA ðxÞ < a and mA ðxÞ > bg;
are called ða; bÞ-lower cut set and ða; bÞ-strong lower cut set of A respectively [30]. 3. The extension from IFSS to IFS Let FPðXÞ ¼ fA j A ¼ hX; A1 ; A2 i is an IFSS over X}, we first give the definition of intuitionistic nested set. Definition 3.1. If mapping H : ½0; 1 ! FPðXÞ satisfies:
k1 < k2 ) Hðk2 Þ Hðk1 Þ; then H is called an intuitionistic nested set (INS) over X. Note 3.1. Let HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi. Then we have
ð1Þ Hl ðkÞ \ Hm ðkÞ ¼ ;; ð2Þ k1 < k2 ) Hl ðk1 Þ Hl ðk2 Þ; Hm ðk1 Þ Hm ðk2 Þ: Example 3.1. Let X ¼ fx1 ; x2 ; x3 ; x4 g and
8 hX; fx1 ; x2 ; x3 ; x4 g; ;i; > > > < hX; fx ; x g; fx gi; 2 3 4 HðkÞ ¼ > hX; fx3 g; fx4 gi; > > : hX; ;; fx1 ; x4 gi;
0 6 k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 1
Then we have that
ðaÞ Hl ðkÞ \ Hm ðkÞ ¼ ;; ðbÞ k1 < k2 ) Hl ðk1 Þ Hl ðk2 Þ; Hm ðk1 Þ Hm ðk2 Þ: Thus H is an INS over X. Let UPðXÞ ¼ fH j H is an INS over Xg. We define the operations in UPðXÞ as follows:
[ Hc :
c2C
c2C
\ Hc :
c2C
[ Hc ðkÞ ¼ [ Hc ðkÞ; i:e:; [ Hc ðkÞ ¼ [ ðHc Þl ðkÞ; [ Hc ðkÞ ¼ \ ðHc Þm ðkÞ c2C
c2C
c2C
l
c2C
m
c2C
\ Hc ðkÞ ¼ \ Hc ðkÞ; i:e:; \ Hc ðkÞ ¼ \ ðHc Þl ðkÞ; \ Hc ðkÞ ¼ [ ðHc Þm ðkÞHc : ðHc ÞðkÞ
c2C
c2C
c
c2C
c
c2C
l
c2C
m
c2C
c
¼ ðHð1 kÞÞ ; i:e:; ðH Þl ðkÞ ¼ Hm ð1 kÞ; ðH Þm ðkÞ ¼ Hl ð1 kÞ: Then we have the following conclusion. Theorem 3.1. ðUPðXÞ; [; \; cÞ is a complete De Morgan algebra, i.e., ðUPðXÞ; [; \; cÞ is a complete lattice and c
ð1Þ ðHc Þ ¼ H; c c \ Hc ¼ [ ðHc Þc ; ð2Þ [ Hc ¼ \ ðHc Þc ; c2C c2C c2C c2C ð3Þ H \ [ Hc ¼ [ ðH \ Hc Þ; H [ \ Hc ¼ \ ðH [ Hc Þ: c2C
c2C
c2C
c2C
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Clearly, if we set
H1 H2 () H1 ðkÞ H2 ðkÞ; 8k 2 ½0; 1; then is a partial order and H1 H2 iff H1 [ H2 ¼ H2 iff H1 \ H2 ¼ H1 . If we define the relation \ " in UPðXÞ as
H1 H2 () \ H1 ðaÞ ¼ \ H2 ðaÞ; 8k 2 ð0; 1; a
a
i.e.,
H1 H2 () \ ðH1 Þl ðaÞ ¼ \ ðH2 Þl ðaÞ; [ ðH1 Þm ðaÞ ¼ [ ðH2 Þm ðaÞ; a
a
a
a
then is an equivalent relation over UPðXÞ. For H 2 UPðXÞ; we let ½H ¼ fH0 j H0 2 UPðXÞ; H0 Hg and IF 0 ðXÞ ¼ f½H j H 2 UPðXÞg. Let
F H : ½0; 1 ! FPðXÞ k # F H ðkÞ ¼ \ HðaÞ; F H ð0Þ ¼ X ¼ hX; X; ;i; a
F H : ½0; 1 ! FPðXÞ k # F H ðkÞ ¼ [ HðaÞ; F H ð1Þ ¼ ; ¼ hX; ;; Xi: a>k
Then we have the following conclusions. Lemma 3.1. (1) (2) (3) (4) (5)
F H ; F H 2 UPðXÞ; k1 < k2 ) F H ðk1 Þ F H ðk2 Þ; FH H FH; F H ðkÞ ¼ [a>k F H ðaÞ; 8k 2 ½0; 1Þ; F H ðkÞ ¼ \a
Proof. (1) is obvious. (2) Let k1 < k2 and a0 2 ð0; 1Þ such that k1 < a0 < k2 . Then we have that
F H ðk1 Þ ¼ [ HðaÞ Hða0 Þ \ HðaÞ ¼ F H ðk2 Þ: a>k1
a
(3) By HðaÞ HðkÞ; 8a > k, we have that F H ðkÞ ¼ [a>k HðaÞ HðkÞ. Similarly, we have that F H ðkÞ ¼ \a
H H0 () [ HðaÞ ¼ [ H0 ðaÞ; 8k 2 ½0; 1Þ: a>k
a>k
Proof. We only need to prove F H ðaÞ ¼ F H0 ðaÞ; 8a 2 ð0; 1. Then
that
8a 2 ð0; 1; F H ðaÞ ¼ F H0 ðaÞ () 8a 2 ½0; 1Þ; F H ðaÞ ¼ F H0 ðaÞ.
In
fact,
Let
F H ðkÞ ¼ [ F H ðaÞ ¼ [ F H0 ðaÞ ¼ F H0 ðkÞ; 8k 2 ½0; 1Þ: a>k
a>k
On the other hand, let F H ðaÞ ¼ F 0H ðaÞ; 8a 2 ð0; 1. Then
F H ðkÞ ¼ \ F H ðaÞ ¼ \ F H0 ðaÞ ¼ F H0 ðkÞ; 8k 2 ð0; 1: a
a
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Lemma 3.3. Let Hc H0c ; H H0 ðc 2 CÞ. Then
\ Hc \ H0c ; [ Hc [ H0c ; Hc H0c :
c2C
c2C
c2C
c2C
Proof. (1) Let Hc H0c ðc 2 CÞ, i.e., \a
\
\ Hc ðaÞ ¼ \ \ Hc ðaÞ ¼ \ \ Hc ðaÞ ¼ \ \ H0c ðaÞ ¼ \ \ H0c ðaÞ:
a
a
c2Ca
c2Ca
a
0
It follows that \c2C Hc \c2C Hc . (2) Let Hc H0c ðc 2 CÞ, i.e., [a>k Hc ðaÞ ¼ [a>k H0c ðaÞ; 8k 2 ½0; 1Þ. Then
[
[ Hc ðaÞ ¼ [ [ Hc ðaÞ ¼ [ [ Hc ðaÞ ¼ [ [ H0c ðaÞ ¼ [ [ H0c ðaÞ:
a>k c2C
a>kc2C
c2Ca>k
c2Ca>k
a>k c2C
0
It follows that [c2C Hc [c2C Hc . (3) Let H H0 ; i:e:; [a>k HðaÞ ¼ [a>k HðaÞ; 8k 2 ½0; 1Þ. Then c
c
c
\a
[ ½Hc ¼
c2C
[ Hc ; \ ½Hc ¼ \ Hc ; ½Hc ¼ ½Hc :
c2C
c2C
c2C
By Lemma 3.3, we know that Definition 3.2 is well-defined. Let A ¼ hX; A1 ; A2 i 2 FPðXÞ; HA ðkÞ hX; A1 ; A2 i; 8k 2 ½0; 1 and FP 0 ðXÞ ¼ f½HA j A 2 FPðXÞg. Then ðFP 0 ðXÞ; [; \; cÞ is isomorphic with ðFPðXÞ; [; \; cÞ and ðFP 0 ðXÞ IF 0 ðXÞ. Let IF ðXÞ ¼ ðIF 0 ðXÞ FP0ðXÞÞ [ FPðXÞ, then we obtain a new algebra system ðIF ðXÞ; [; \; cÞ, and ðIF ðXÞ; [; \; cÞ is isomorphic with system ðIF 0 ðXÞ; [; \; cÞ by embedding theorem of algebra [24]. Theorem 3.2. Let f : IF ðXÞ ! LX ,
A ¼ ½H # hX; lA ; mA i; where
lA ðxÞ ¼ _fk j x 2 Hl ðkÞg; mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg. Then f is a surjection.
Proof. Firstly, we show that f is well-defined. Let A ¼ ½H ¼ ½H0 . Then F H ðkÞ ¼ F H0 ðkÞ H0 ðkÞ F H0 ðkÞ ¼ F H ðkÞ; 8k 2 ½0; 1, i.e.,
ðF H Þl ðkÞ ¼ ðF H0 Þl ðkÞ ðH0 Þl ðkÞ ðF H0 Þl ðkÞ ¼ ðF H Þl ðkÞ; ðF H Þm ðkÞ ¼ ðF H0 Þm ðkÞ ðH0 Þm ðkÞ ðF H0 Þm ðkÞ ¼ ðF H Þm ðkÞ: Thus
_fk j x 2 ðF H Þl ðkÞg 6 _fk j x 2 ðH0 Þl ðkÞg 6 fk j x 2 ðF H Þl ðkÞg: Let d 2 fk j x 2 ðF H Þl ðkÞg: Then x 2 ðF H Þl ðdÞ. By Lemma 3.1(2) we have that x 2 ðF H Þl ðdÞ ðF H Þl ðkÞ; 8k < d. It follows that _fk j x 2 ðF H Þl ðkÞg P _fk j k < dg ¼ d. Thus _fk j x 2 ðF H Þl ðkÞg P _fd j x 2 ðF H Þl ðdÞg. Then
_fk j x 2 ðF H Þl ðkÞg ¼ _fk j x 2 ðH0 Þl ðkÞg ¼ _fk j x 2 ðF H Þl ðkÞg: Therefore, lA ðxÞ ¼ _fk j x 2 Hl ðkÞg ¼ _fk j x 2 ðH0 Þl ðkÞg. Similarly,
^f1 k j x R ðF H Þm ðkÞg P ^f1 k j x R ðH0 Þm ðkÞg P ^f1 k j x R ðF H Þm ðkÞg Let d 2 fk j x R ðF H Þm ðkÞg. By Lemma 3.1(2), we have that x R ðF H Þm ðdÞ ðF H Þm ðkÞ; 8k < d, and consequently x R ðF H Þm ðkÞ; 8k < d. Then ^f1 k j x R ððFÞH Þm ðkÞg 6 ^f1 k j k < dg ¼ 1 d. Thus ^f1 k j x R ðF H Þm ðkÞg 6 ^f1 d j x R ðF H Þm ðdÞg. It follows that
^f1 k j x R ðF H Þm ðkÞg ¼ ^f1 k j x R ðH0 Þm ðkÞg ¼ ^f1 k j x R ðF H Þm ðkÞg:
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Thus
mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg ¼ ^f1 k j x R ðH0 Þm ðkÞg: Therefore, f is well-defined. Secondly, we prove that lA ðxÞ þ mA ðxÞ 6 1; 8x 2 X. In fact, assume that
lA ðxÞ þ mA ðxÞ > 1, then we have that
_fk j x 2 Hl ðkÞg > 1 ^f1 k j x R Hm ðkÞg ¼ _fk j x R Hm ðkÞg: Thus there exists a k0 2 ½0; 1 such that x 2 Hl ðk0 Þ and k0 > _fk j x R Hm ðkÞg. From Hl ðk0 Þ \ Hm ðk0 Þ ¼ ;, we have x0 R Hm ðk0 Þ. It follows that k0 > _fk j x R Hm ðkÞg P k0 , which leads to a contradiction. Finally, we show that f is a surjection. Let hX; lA ; mA i 2 LX and ðHA ÞðkÞ ¼ hX; ðHA Þl ðkÞ; ðHA Þm ðkÞi, where ðHA Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg and ðHA Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg; 8k 2 ½0; 1. Then we have that HA 2 UPðXÞ and _fk j x 2 ðHA Þl ðkÞg ¼ lA ðxÞ; ^f1 k j x R ðHA Þm ðkÞg ¼ mA ðxÞ. Thus f ð½HA Þ ¼ hX; lA ; mA i. Therefore f is a surjection. h Theorem 3.3. Let A ¼ ½H 2 IF ðXÞ and hX; lA ; mA i ¼ f ð½HÞ. Then
ð1Þ ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg; ð2Þ ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg; ð3Þ ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ > kg; ð4Þ ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ P 1 kg:
Proof. (1) If x 2 ðF H Þl ðkÞ, then we have that lA ðxÞ ¼ _fa j x 2 ðF H Þl ðaÞg P k. On the other hand, if x R ðF H Þl ðkÞ ¼ \a
mA ðxÞ ¼ ^f1 a j x R ðHÞm ðaÞg P ^f1 a j a 6 a0 g ¼ 1 a0 > 1 k: On the other hand, x R ðF H Þm ðkÞ ) x R ðF H Þm ðaÞ; 8a < k ) mA ðxÞ ¼ ^f1 a j x R ðF H Þm ðaÞg 6 ^f1 a j a < kg ¼ 1 k. It follows that ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg. Proofs of (3) and (4) are similar.
h
Theorem 3.4. ðIF ðXÞ; [; \; cÞffif ðLX ; _; ^; cÞ; i.e., f is a bijection and satisfies that
\ ½Hc ¼ \ f ð½Hc Þ; c2C c2C ð2Þ f [ ½Hc ¼ [ f ð½Hc Þ;
ð1Þ f
c2C
c2C
c
ð3Þ f ð½Hc Þ ¼ ðf ð½HÞÞ : Proof. (1) Let f ð½HÞ ¼ f ð½H0 Þ ¼ hX; lA ; mA i. Then
ðF H Þl ðkÞ ¼ fx j x 2 X; lA ðxÞ P kg ¼ ðF H0 Þl ðkÞ; 8k 2 ½0; 1; ðF H Þm ðkÞ ¼ fx j x 2 X; mA ðxÞ > 1 kg ¼ ðF H0 Þm ðkÞ; 8k 2 ½0; 1: Thus F H ðkÞ ¼ F H0 ðkÞ; 8k 2 ½0; 1 and consequently H H0 , i.e., ½H ¼ ½H0 . Therefore f is a bijection. (2) Let AðcÞ ¼ ½Hc 2 IF ðXÞ; f ðAðcÞ Þ ¼ hX; lAðcÞ ; mAðcÞ i and ½H 2 IF ðXÞ such that f ½H ¼ \c2C hX; lAðcÞ ; mAðcÞ i ¼ hX; \c2C lAðcÞ ; [c2C mAðcÞ i. Then
ðF H Þl ðkÞ ¼ x j x 2 X; ^c2C lAðcÞ ðxÞ P k ¼ \ fx j x 2 X; lAðcÞ ðxÞ P kg c2C ¼ \ ðF HðcÞ Þl ðkÞ ¼ \ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C c2C l ðF H Þm ðkÞ ¼ x j x 2 X; _c2C mAðcÞ ðxÞ > 1 k ¼ [ fx j x 2 X; mAðcÞ ðxÞ > 1 kg c2C ¼ [ ðF HðcÞ Þm ðkÞ ¼ \ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
m
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
7
Thus ðF H ÞðkÞ ¼ ð\c2C F HðcÞ ÞðkÞ; 8k 2 ½0; 1. It follows that
½H ¼ ½F H ¼
\ F HðcÞ ¼ \ ½F HðcÞ ¼ \ ½HðcÞ ¼ \ AðcÞ :
c2C
c2C
c2C
c2C
and
f
\ AðcÞ
c2C
ðcÞ
(3) Let A
¼ f ð½HÞ ¼ \ hX; lAðcÞ ; mAðcÞ i ¼ \ f ðAðcÞ Þ: c2C
c2C
ðcÞ
¼ ½HðcÞ 2 IF ðXÞ; f ðA Þ ¼ hX; lAðcÞ ; mAðcÞ i and ½H 2 IF ðXÞ such that
f ð½HÞ ¼ [ f ðAðcÞ Þ ¼ c2C
X; [ lAðcÞ ; \ mAðcÞ : c2C
c2C
Then
ðF H Þl ðkÞ ¼ fx j x 2 X; _c2C lAðcÞ ðxÞ > kg ¼ [ fx j x 2 X; lAðcÞ ðxÞ > kg c2C ¼ [ ðF HðcÞ Þl ðkÞ ¼ [ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
l
ðF H Þm ðkÞ ¼ fx j x 2 X; ^c2C mAðcÞ ðxÞ P 1 kg ¼ \ fx j x 2 X; mAðcÞ ðxÞ P 1 kg c2C ¼ \ ðF HðcÞ Þm ðkÞ ¼ [ F HðcÞ ðkÞ; 8k 2 ½0; 1: c2C
c2C
m
Thus ½H ¼ ½F H ¼ ½[c2C F HðcÞ ¼ [c2C ½F HðcÞ ¼ [c2C ½HðcÞ ¼ [c2C AðcÞ . Therefore, f ð[c2C AðcÞ Þ ¼ [c2C f ðAðcÞ Þ. (4) Let A ¼ ½H 2 IF ðXÞ; f ð½HÞ ¼ hX; lA ; mA i and ½H 2 IF ðXÞ, such that f ð½HÞ ¼ hX; mA ; lA i. Then
ðF H Þl ðkÞ ¼ fx j x 2 X; mA ðxÞ P kg ¼ ðF H Þm ð1 kÞ and
ðF H Þm ðkÞ ¼ fx j x 2 X; lA ðxÞ > 1 kg ¼ ðF H Þl ð1 kÞ: Thus
ðF H ÞðkÞ ¼ hX; ðF H Þm ð1 kÞ; ðF H Þl ð1 kÞi ¼ hX; ðF H Þl ð1 kÞ; ðF H Þm ð1 kÞic ¼ ðF H ð1 kÞÞc ¼ ðF H Þc ðkÞ: It follows that ½H ¼ ½F H ¼ ½ðF H Þc ¼ ½F H c ¼ ½Hc . Therefore, f ð½Hc Þ ¼ f ð½HÞ ¼ f ð½HÞc . h Example 3.2. Let X ¼ fðx; yÞ j x; y 2 ½0; 1; x2 þ 2y2 6 1g. For k 2 ½0; 1, we set
HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi;
Fig. 1.
lA and Hl ðkÞ of k-cut set on IFS ðlA ; mA Þ.
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
Fig. 2.
mA and Hm ðkÞ of k-cut set on IFS ðlA ; mA Þ.
where Hl ðkÞ ¼ fðx; yÞ j x; y 2 X; x2 þ 2y2 6 1 kg; Hm ðkÞ ¼ fðx; yÞ j x; y 2 X; x2 þ 2y2 >
pffiffiffiffiffiffiffiffiffiffiffiffi 1 k. Let A ¼ TðHÞ ¼ hX; lA ; mA i, then
lA ðx; yÞ ¼ _fk j ðx; yÞ 2 Hl ðkÞg ¼ 1 ðx2 þ 2y2 Þ; 2
mA ðx; yÞ ¼ ^f1 k j ðx; yÞ R Hm ðkÞg ¼ ðx2 þ 2y2 Þ : When k ¼ 0, 0.25, 0.5, 0.75, 1, lA and kHl ðkÞ are shown in the Fig. 1(a) and (b) respectively. mA and k Hm ðkÞ are shown in the Fig. 2(a) and (b) respectively. Note 3.2. By Theorem 3.4, we know that an IFS hX; lA ; mA i presented by Atanassov can be seen as an equivalent class ½H of an INS, which is just as a real number can be seen as an equavalent class of rational number. Therefore, we will not distanguish A ¼ ½H from f ð½HÞ ¼ hX; lA ; mA i in the following discussions. 4. The cut sets of IFSs By Theorem 3.3, we can easily give the definition of k-cut set of IFS. Definition 4.1. Let A ¼ hX; lA ; mA i can be an IFS over X; k 2 ½0; 1. Let
Alk ¼ fx j x 2 X; lA ðxÞ P kg;
Am½k ¼ fx j x 2 X; mA ðxÞ > 1 kg;
Alk ¼ fx j x 2 X; lA ðxÞ > kg;
Am½k ¼ fx j x 2 X; mA ðxÞ P 1 kg:
Then Ak ¼ hX; Alk ; Am½k i and Ak ¼ hX; Alk ; Am½k i are called k-cut set and k-strong cut set of the A, respectively. Example 4.1. Let X ¼ fx1 ; x2 ; x3 ; x4 g and A ¼ hX; lA ; mA i, where
lA ¼ ð0:6; 0:4; 0:4; 0:8Þ; mA ¼ ð0:1; 0:2; 0:6; 0:1Þ: Then
8 X; > > > < fx ; x g; 1 4 l Ak ¼ > fx4 g; > > : ;;
Am½k
8 ;; > > > < fx3 g; ¼ > fx > 2 ; x3 g; > : X;
k 6 0:4 0:4 < k 6 0:8 0:6 < k 6 0:8 0:8 < k 6 1
8 X; > > > < fx ; x g; 1 4 l Ak ¼ > fx4 g; > > : ;;
k 6 0:4 0:4 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1
Am½k
8 ;; > > > < fx3 g; ¼ > > fx2 ; x3 g; > : X;
k < 0:4 0:4 6 k < 0:8 0:6 6 k < 0:8 0:8 6 k 6 1 k < 0:4 0:4 6 k < 0:8 0:8 6 k < 0:9 0:9 6 k 6 1
Thus k-cut set and k-strong cut set of the A have the following forms respectively: Please cite this article in press as: X.-h. Yuan et al., The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.044
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
8 hX; X; ;i; > > > > > > > < hX; fx1 ; x4 g; fx3 gi; Ak ¼ hX; fx4 g; fx3 gi; > > > > hX; ;; fx2 ; x3 gi; > > > : hX; ;; Xi; 8 hX; X; ;i; > > > > > > > < hX; fx1 ; x4 g; fx3 gi; Ak ¼ hX; fx4 g; fx3 gi; > > > > hX; ;; fx2 ; x3 gi; > > > : hX; ;; Xi;
k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1 k < 0:4 0:4 6 k < 0:6 0:6 6 k < 0:8 0:8 6 k < 0:9 0:9 6 k 6 1
Clearly, we have the following properties. Property 4.1. (1) Let A 2 LX ; HA ðkÞ ¼ Ak ; HA ðkÞ ¼ Ak , then
HA ; HA 2 UPðXÞ; HA HA ; (2) k1 < k2 ) Ak1 Ak2 ; (3) Let A ¼ hX; lA ; mA i; B ¼ hX; lB ; mB i. Then
ðA \ BÞk ¼ Ak \ Bk ;
ðA \ BÞk ¼ Ak \ Bk ;
ðA [ BÞk ¼ Ak [ Bk ;
ðA [ BÞk ¼ Ak [ Bk ;
X
(4) Let At ; Bt 2 L ðt 2 TÞ. Then
[ At
t2T
\ At
t2T
[ ðAt Þk ; t2T
k
¼ [ ðAt Þk ;
t2T
¼ \ ðAt Þk ; t2T
k
[ At \ At
k
t2T
\ ðAt Þk ;
t2T
k
t2T
(5) ðAc Þk ¼ ðA1k Þc ; ðAc Þk ¼ ðA1k Þc ; (6) Let kt 2 ½0; 1ðt 2 TÞ and a ¼ _t2T kt ; b ¼ ^t2T kt . Then
\ Akt ¼ Aa ; [ Akt ¼ Ab ;
t2T
t2T
In general, Ak ¼ [a>k Aa ¼ [a>k Aa ; Ak ¼ \ Aa ¼ \ Aa ; a
Proof. Firstly, we show A ¼ [t2T At ; B ¼ \ At , then
that
ð[t2T At Þk [t2T ðAt Þk ,
i.e.,ð[t2T At Þlk [t2T ðAt Þlk ; ð[t2T At Þm½k \ ðAt Þm½k . t2T
In
fact,
let
t2T
x 2 [ ðAt Þlk ) 9t0 ; x 2 ðAt0 Þlk ) 9t0 ; t2T
lAt0 ðxÞ P k ) lA ðxÞ ¼ _t2T lAt ðxÞ P k ) x 2 Alk ;
x 2 Am½k ) ^t2T mAt ðxÞ ¼ mA ðxÞ > 1 k ) 8t 2 T; l
l
m
mAt ðxÞ > 1 k ) 8t 2 T; x 2 ðAt Þm½k ) x 2 \ ðAt Þm½k : t2T
m
Then Ak [t2T ðAt Þk ; A½k \t2T ðAt Þ½k . Thus ð[t2T At Þk [t2T ðAt Þk . l Secondly, we show that Aa ¼ \t2T Akt , i.e., Ala ¼ \t2T Akt ; Am½a ¼ [t2T Am½kt . In fact, l
l
x 2 \ Akt () 8t 2 T; x 2 Akt () 8t 2 T; t2T
x 2 [ Am½kt () 9t 0 ; t2T
lA ðxÞ P kt () lA ðxÞ P _kt ¼ a () x 2 Ala ;
mA ðxÞ > 1 kt0 () mA ðxÞ > ^t2T ð1 kt Þ ¼ 1 _t2T kt ¼ 1 a () x 2 Am½a :
l Then Ala ¼ \t2T Akt ; Am½a ¼ [ Am½kt . Thus Aa ¼ \t2T Akt . t2T
l
m
l
Finally, we show that ðAc Þk ¼ ðA1k Þc ; i:e:; ðAc Þk ¼ Am½1k ; ðAc Þ½k ¼ A1k . In fact,
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l
x 2 ðAc Þk () mA ðxÞ ¼ lAc ðxÞ P k ¼ 1 ð1 kÞ () x 2 Am½1k ; m
l
x 2 ðAc Þ½k () lA ðxÞ ¼ mAc ðxÞ > 1 k () x 2 A1k : l
m
l
Then ðAc Þk ¼ Am½1k ; ðAc Þ½k ¼ A1k . Thus ðAc Þk ¼ ðA1k Þc . Others are obvious. h Example 4.2. Let X ¼ fa; bg; AðaÞ ¼ ð0:4; 0:4Þ; AðbÞ ¼ ð0:4; 0:3Þ; BðaÞ ¼ ð0:4; 0:3Þ; BðbÞ ¼ ð0:5; 0:4Þ; CðaÞ ¼ ð0:5; 0:4Þ; CðbÞ ¼ ð0:4; 0:3Þ, ða1 ; b1 Þ ¼ ð0:4; 0:4Þ; ða2 ; b2 Þ ¼ ð0:3; 0:3Þ, ðk1 ; k2 Þ ¼ ð0:4; 0:4Þ. Then Ac ðaÞ ¼ ð0:4; 0:4Þ, Ac ðbÞ ¼ ð0:3; 0:4Þ; ðB [ CÞðaÞ ¼ ðB [ CÞðbÞ ¼ ð0:5; 0:3Þ, ða; bÞ ¼ ða1 ; b1 Þ ^ ða2 ; b2 Þ ¼ ð0:3; 0:4Þ; ðk1 ; k2 Þc ¼ ð0:4; 0:4Þ. By Definition 2.7, we have that ðAc Þðk1 ;k2 Þ ¼ fag; Aðk1 ;k2 Þc ¼ Að0:4;0:4Þ ¼ ;, ðB [ CÞðk1 ;k2 Þ ¼ fa; bg; Bðk1 ;k2 Þ ¼ C ðk1 ;k2 Þ ¼ ;, Aða1 ;b1 Þ ¼ Aða2 ;b2 Þ ¼ ;; Aða;bÞ ¼ fbg. Then
ðAc Þðk1 ;k2 Þ – ðAðk1 ;k2 Þc Þc ; ðB [ CÞðk1 ;k2 Þ – Bðk1 ;k2 Þ [ C ðk1 ;k2 Þ ; Aða1 ;b1 Þ [ Aða2 ;b2 Þ – Aða;bÞ : Therefore, ða; bÞ-upper cut sets of IFS presented in [15] do not satisfy the Property 4.1(3, 4, 5, 6). Similarly, we can show that ða; bÞ-lower cut sets of IFS presented in [30] do not also satisfy the Property 4.1(3, 4, 5, 6). Therefore, k-cut sets presented by us have more properties than ða; bÞ-cut sets in [15,30]. Note 4.1. By comparison with the properties of cut sets on Zadeh fuzzy sets in [18], we have known that cut sets of IFSs have the same properties as cut sets of Zadeh’s fuzzy sets.
5. Decomposition theorem, representation theorem and extension principle
Definition 5.1. Let A ¼ hX; A1 ; A2 i 2 FPðXÞ; k 2 ½0; 1. Then
ðkA1 ÞðxÞ ¼
k;
x 2 A1
0; x R A1
ðk A2 Þ ¼
1 k; x R A2 1;
x 2 A2
kA ¼ hX; kA1 ; k A2 i:
Property 5.1. A B ) kA kB; i:e:; hX; kA1 ; k A2 i hX; kB1 ; k B2 i Clearly, we have the following theorem from the discussion in Section 3. Theorem 5.1 (Decomposition Theorem). Let A ¼ hX; lA ; mA i be an IFS and Ak and Ak are k-cut set and k-strong cut set of A respectively. Let H 2 FPðXÞ such that Ak HðkÞ Ak . Then
ð1Þ A ¼ [ kAk ; k2½0;1
ð2Þ A ¼ [ kAk ; k2½0;1
ð3Þ A ¼ [ kHðkÞ and k2½0;1
ðaÞ k1 < k2 ) Hðk1 Þ Hðk2 Þ; ðbÞ Ak ¼ \ HðaÞ; and a
ðcÞ Ak ¼ [ HðaÞ: a>k
l
Proof. Let B ¼ [k2½0;1 kAk ¼ hX; [k2½0;1 kAk ; \k2½0;1 k Am½k i. Then
lB ðxÞ ¼ mB ðxÞ ¼
l l [ kAk ðxÞ ¼ _fk j x 2 Ak g ¼ _fk j lA ðxÞ P kg ¼ lA ðxÞ;
k2½0;1
n o l l \ k A½k ðxÞ ¼ ^ 1 k j x R A½k ¼ ^f1 k j mA ðxÞ 6 1 kg ¼ mA ðxÞ:
k2½0;1
Then A ¼ B ¼ [k2½0;1 kAk . (2) And (3) are obvious. h
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Fig. 3.
lA and Hl ðkÞ in Example 4.1.
Fig. 4.
mA and Hm ðkÞ in Example 4.1.
Note 5.1. By Theorem 5.1(3), we have know that A ¼ [k2½0;1 kHðkÞ means lA ðxÞ ¼ _fk j x 2 Hl ðkÞg and x R Hm ðkÞg. Therefore, decomposition theorem has built the connections between IFS and IFSS.
11
mA ðxÞ ¼ ^f1 k j
Example 5.1. In Example 4.1, we have that
8 ðk; k; k; kÞ; > > > < ðk; 0; 0; kÞ; kAlk ¼ > ð0; 0; 0; kÞ; > > : ð0; 0; 0; 0Þ;
m
k A½k
k 6 0:4 0:4 < k 6 0:6 0:6 < k 6 0:8 0:8 < k 6 1
8 ð1 k; 1 k; 1 k; 1 kÞ; > > > < ð1 k; 1 k; 1; 1 kÞ; ¼ > ð1 k; 1; 1; 1 kÞ; > > : ð1; 1; 1; 1Þ;
k 6 0:4 0:4 < k 6 0:8 0:8 < k 6 0:9 0:9 < k 6 1
l
Then [k2½0;1 kAk ¼ ð_k60:4 ðk; k; k; kÞÞ _ ð_0:4
\k2½0;1 k Amk ¼ ð^k60:4 ð1 k;1 k;1 k;1 kÞÞ ^ ð^0:4
T : UPðXÞ ! LX ; H # TðHÞ ¼
X; [ kHl ðkÞ; \ k Hm ðkÞ k2½0;1
k2½0;1
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Then T is a homomorphism from ðUPðXÞ; [; \; cÞ onto ðLX ; [; \; cÞ, i.e., f is a surjection and
ð1Þ T
[ Hc
c2C
¼ [ TðHc Þ; T
c2C
\ Hc
c2C
¼ \c2C TðHc Þ; TðHc Þ ¼ ðTðHÞÞc ;
ð2Þ ðTðHÞÞk HðkÞ ðTðHÞÞk ; ð3Þ ðTðHÞÞk ¼ \ HðaÞ; ðTðHÞÞk ¼ [ HðaÞ: a
a>k
Note 5.2. Representation theorem has shown that each INS can construct an IFS and each IFS can be constructed by an INS. Example 5.2. Let X ¼ ½0; 2. For k 2 ½0; 1, we set HðkÞ ¼ hX; Hl ðkÞ; Hm ðkÞi, h pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i Hm ðkÞ ¼ 0; 1 1 k [ 1 þ 1 k; 2 . Then H is an INS over X. Thus
where
Hl ðkÞ ¼
hpffiffiffi pffiffiffii k; 2 k
and
(a) For x 2 ½0; 1, we have that
n
o
pffiffiffi
lA ðxÞ ¼ _fk j x 2 Hl ðkÞg ¼ _ k j k 6 x 6 1 ¼ _fk j k 6 x2 g ¼ x2 : n
pffiffiffiffiffiffiffiffiffiffiffiffi
mA ðxÞ ¼ ^f1 k j x R Hm ðkÞg ¼ ^ 1 k j 1 1 k 6 x 6 1
o
n pffiffiffiffiffiffiffiffiffiffiffiffio ¼^ 1kj061x6 1k
¼ ^f1 k j ð1 xÞ2 6 1 kg ¼ ð1 xÞ2 : (b) For x 2 ð1; 2, we have that
pffiffiffiffiffi
lA ðxÞ ¼ _fk j 1 < x 6 2 kg ¼ _fk j k 6 ð2 xÞ2 g ¼ ð2 xÞ2 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi mA ðxÞ ¼ ^f1 k j 1 < x 6 1 þ 1 kg ¼ ^f1 k j 0 < x 1 6
pffiffiffiffiffiffiffiffiffiffiffiffi 1 kg
¼ ^f1 k j ðx 1Þ2 6 1 kg ¼ ðx 1Þ2 : Therefore,
2 lA ðxÞ ¼ x ;
ð2 xÞ2 ;
06x61 1
mA ðxÞ ¼ ðx 1Þ2 .
When k ¼ 0, 0.25, 0.5, 0.75, 1, lA and kHl ðkÞ are represented in the Fig. 3(a) and (b), respectively. represented in the Fig. 4(a) and (b), respectively.
mA and k Hm ðkÞ are
Example 5.3. Let X ¼ fa; bg; AðaÞ ¼ ð0:4; 0:4Þ; AðbÞ ¼ ð0:4; 0:3Þ; a; b 2 ½0; 1 and a þ b 6 1, we set
Hða; bÞ ¼ A
ða;bÞ
8 > < fa; bg; a P 0:4; b 6 0:3; ¼ fag; a P 0:4; 0:3 < b 6 0:4; > : ;; else:
By the operation defined in [30], we have that
8 > < ;; b P 0:4; a 6 0:3; Hc ða; bÞ ¼ ðHðb; aÞÞc ¼ fbg; b P 0:4; 0:3 < a 6 0:4; > : fa; bg; else: By the representation theorem in [30], we have that
TðHÞðxÞ ¼ ^fða; bÞ j a; b 2 ½0; 1; a þ b 6 1; x 2 Hða; bÞg: Then it follows that TðHÞðaÞ ¼ AðaÞ ¼ ð0:4; 0:4Þ, ðTðHc ÞÞðaÞ ¼ ð0; 0:6Þ – ðTðHÞÞc ðaÞ. Thus the representation theorem in [30] does not preserve the operation c. Therefore, the representation theorem presented by us has better properties than that in [30]. By using Theorem 5.2 (representation theorem) and Definition 2.4, we can easily establish extension principle of IFS. Theorem 5.3 (Extension Principle). Let f : X ! Y be a mapping. Let
ðaÞ f : LX ! LY A ¼ hX; lA ; mA i # f ðAÞ ¼ [ kf ðAk Þ k2½0;1 l ¼ Y; [ kf ðAk Þ; \ k f ðAm½k Þ : k2½0;1
k2½0;1
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ðbÞ f 1 : LY ! LX B ¼ hY; lB ; mB i # f 1 ðBÞ ¼ [ kf 1 ðBk Þ k2½0;1 l 1 ¼ X; [ kf ðBk Þ; \ k f 1 ðBm½k Þ : k2½0;1
k2½0;1
Then for any A1 ; A2 ; At ðt 2 TÞ; A 2 LX and B1 ; B2 ; Bt ðt 2 TÞ; B 2 LY , we have that
lf ðAÞ ðyÞ ¼ _f ðxÞ¼y lA ðxÞ; mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ; ð2Þ lf 1 ðBÞ ðxÞ ¼ lB ðf ðxÞÞ; mf 1 ðBÞ ðxÞ ¼ mB ðf ðxÞÞ;
ð1Þ
ð3Þ A1 A2 ) f ðA1 Þ f ðA2 Þ; ð4Þ B1 B2 ) f 1 ðB1 Þ f 1 ðB2 Þ; ð5Þ A f 1 ðf ðAÞÞ; f ðf 1 ðBÞÞ B; c ð6Þ f 1 [ Bt ¼ [ f 1 ðBt Þ; f 1 \ Bt ¼ \ f 1 ðBt Þ; f 1 ðBc Þ ¼ ðf 1 ðBÞÞ ; t2T t2T t2T t2T ð7Þ f [ At ¼ [ f ðAt Þ; f \ At \ f ðAt Þ; f ðAÞc f ðAc Þ; t2T
t2T
t2T
t2T
ð8Þ ðf ðAÞÞk ¼ f ðAk Þ; ðf ðAÞÞk f ðAk Þ ð9Þ ðf 1 ðBÞÞk ¼ f 1 ðBk Þ; ðf 1 ðBÞÞk ¼ f 1 ðBk Þ:
mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ. In fact, let f 1 ðyÞ ¼ fx j x 2 X; f ðxÞ ¼ yg. Then
Proof. Firstly, we show that
n
n c o n c o ¼ ^ 1 k j y 2 f Am½k ¼ 1 k j 9x 2 Am½k ; f ðxÞ ¼ y n o ¼ ^ 1 k j 9x R Am½k ; f ðxÞ ¼ y ¼ ^f1 k j 9x 2 X; f ðxÞ ¼ y; mA ðxÞ 6 1 kg:
mf ðAÞ ðyÞ ¼ ^ 1 k j y R f Am½k
o
Thus, we can easily show that mf ðAÞ ðyÞ ¼ ^f ðxÞ¼y mA ðxÞ. Secondly, we show that ðf ðAÞÞk ¼ f ðAk Þ; ðf ðAÞÞk f ðAk Þ. In fact,
ðf ðAÞÞk ¼ [ f ðAa Þ ¼ f a>k
ðf ðAÞÞk ¼ \ f ðAa Þ f
¼ f ðAk Þ;
[ Aa
a>k
a
\ Aa
a
¼ f ðAk Þ:
Others are obvious. h l
Example 5.4. Let X ¼ ½0; 2; Y ¼ ½0; 4 and f : X ! Y; x # x2 . In the Example 5.2, k-cut set of the A is Ak ¼ hX; Ak ; Am½k i, where hpffiffiffi h c c pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffii pffiffiffi2 l l Ak ¼ and f Am½k ¼ f Am½k k; 2 k and Am½k ¼ 0; 1 1 k [ 1 þ 1 k; 2 . Then f ðAk Þ ¼ k; 2 k pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffiffiffiffiffiffi2 c pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1 1k ; 1þ 1k ¼ 0; 1 1 k [ 1 þ 1 k ; 4 . Let B ¼ f ðAÞ ¼ hY; lB ; mB i, then lB ðyÞ ¼ n o l _ k j y 2 f Ak ; mB ðyÞ ¼ ^ 1 k j y R f Am½k . Thus pffiffiffiffiffiffiffiffiffiffiffiffi2 (a) When y 2 ½0; 1, we have that l ðyÞ ¼ _fk j k 6 yg ¼ y and m ðyÞ ¼ ^ 1 k j 1 6 y 1 k B B n pffiffiffi 2 o pffiffiffi 2 ¼ 1 y . ¼^ 1kj1kP 1 y n pffiffiffi2 pffiffiffi 2 o pffiffiffi 2 (b) When y 2 ð1; 4, we have that lB ðyÞ ¼ _ k j y 6 2 k ¼_ kjk6 2 y and mB ðyÞ ¼ ¼ 2 y n pffiffiffiffiffiffiffiffiffiffiffiffi2 pffiffiffi
2 o pffiffiffi
2 y1 y1 . ¼^ 1kj1kP ^ 1kj1
y; lB ðyÞ ¼
2
pffiffiffi 2 y ;
06y61 ; 1
mB ðyÞ ¼
pffiffiffi
2 y1 .
6. An application to intuitionistic fuzzy algebra In this section, we will present another definition of intuitionistic fuzzy subgroup by using the representation theorem (Theorem 5.2). Let G be a group and
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X.-h. Yuan et al. / Information Sciences xxx (2014) xxx–xxx
SPðGÞ ¼ fhG; A1 ; A2 i j A1 \ A2 ¼ ;; A1
and ðA2 Þc are subgroup of Gg:
For our convenience, the empty set ; is also considered as a subgroup of G. Definition 6.1. If mapping H : ½0; 1 ! SPðGÞ k # HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi is an INS, then H is called a GINS over G. Definition 6.2. Let A ¼ hG; lA ; mA i is an IFS of G. If there is a GINS H over G such that A ¼ [k2½0;1 kHðkÞ ¼ hG; [k2½0;1 kHl ðkÞ; \k2½0;1 k Hm ðkÞi, then A is called an intuitionistic fuzzy subgroup of G. Theorem 6.1. An IFS A of G is an intuitionistic fuzzy subgroup of G if and only if for any x; y 2 G
lA ðxyÞ P lA ðxÞ ^ lA ðyÞ; lA ðx1 Þ P lA ðxÞ ðiiÞ mA ðxyÞ 6 mA ðxÞ _ mA ðyÞ; mA ðx1 Þ 6 mA ðxÞ
ðiÞ
ð6:1Þ ð6:2Þ
Proof. Let A ¼ [k2½0;1 kHðkÞ ¼ hG; [k2½0;1 kHl ðkÞ; \k2½0;1 k Hm ðkÞi is an intuitionistic fuzzy subgroup of G. We only need to show that mA ðxyÞ 6 mA ðxÞ _ mA ðyÞ. Since ðHm ðkÞÞc is a subgroup of G, then
f1 k j xy 2 ðHm ðkÞÞc g f1 k j x 2 ðHm ðkÞÞc g \ f1 k j y 2 ðHm ðkÞÞc g;
8x; y 2 G:
Thus
mA ðxyÞ ¼ ^f1 k j k 2 ½0; 1; xy R Hm ðkÞg ¼ ^f1 k j k 2 ½0; 1; xy 2 ðHm ðkÞÞc g 6 ^f1 k j k 2 ½0; 1; x 2 ðHm ðkÞÞc ; y 2 ðHm ðkÞÞc g ¼ ^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g: We need to show that
^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g 6 ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ: In fact, assume that there exists a a 2 ½0; 1 such that
^fð1 kÞ _ Hm ðkÞðxÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g > a > ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ ð Þ Then Hm ð1 aÞðxÞ _ Hm ð1 aÞðyÞ ¼ 1. Thus x 2 Hm ð1 aÞ or y 2 Hm ð1 aÞ. When x 2 Hm ð1 aÞ, we have that x 2 Hm ðkÞ for any k P 1 a and x R Hm ðkÞ implies k < 1 a. Then. ^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1g ¼ ^f1 k j k 2 ½0; 1; x R Hm ðkÞg P ^f1 k j k < 1 ag ¼ a. Similarly, when y 2 Hm ð1 aÞ, we have that ^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1g P a. Then. ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ P a. This contradicts with the formula (⁄). Hence
mA ðxyÞ 6 ð^fð1 kÞ _ Hm ðkÞðxÞ j k 2 ½0; 1gÞ _ ð^fð1 kÞ _ Hm ðkÞðyÞ j k 2 ½0; 1gÞ ¼ ð^f1 k j k 2 ½0; 1; x R Hm ðkÞgÞ _ ð^f1 k j k 2 ½0; 1; y R Hm ðkÞgÞ ¼ mA ðxÞ _ mA ðyÞ: On the other hand, let A ¼ hG; lA ; mA i satisfy the conditions (i) and (ii). Let Hl ðkÞ ¼ fx j x 2 G; lA ðxÞ P kg, Hm ðkÞ ¼ fx j x 2 G; mA ðxÞ > 1 kg and HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi. Then H : ½0; 1 ! SPðGÞk # HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi is a GINS of G and
A ¼ [ kHðkÞ ¼ k2½0;1
G; [ kHl ðkÞ; \ k Hm ðkÞ : k2½0;1
k2½0;1
Therefore, A is an intuitionistic fuzzy subgroup of G. h 2
Example 6.1. Let G ¼ fe; a; b; abg be a group, where e is an identity and ab ¼ ba; a2 ¼ b ¼ e. Let HðkÞ ¼ hG; Hl ðkÞ; Hm ðkÞi, where
8 0 6 k < 0:4 > < G; Hl ðkÞ ¼ fe; ag; 0:4 6 k < 0:8 > : feg; 0:8 < k 6 1 Then ðHm ðkÞÞc ¼
G; fe; ag;
Hm ðkÞ ¼
;;
0 6 k < 0:4
fb; abg; 0:4 6 k 6 1
0 6 k < 0:4 . 0:4 6 k 6 1
Thus Hl ðkÞandðHm ðkÞÞc are subgroup of G and consequently H is a GINS over G. Let A ¼ [k2½0;1 kHðkÞ ¼ hG; lA ; mA i, then we have that lA ¼ ð1; 0:8; 0:4; 0:4Þ and mA ¼ ð0; 0; 0:6; 0:6Þ. Clearly, lA and mA satisfy the formulae 6.1 and 6.2.
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7. Conclusion In this paper, a theoretical base of IFS based on the concept of IFSS was established. Firstly, we showed that an IFS was an equivalent class of an INS. Thus an IFS can be determined by an INS. We also presented the concept of k-cut set of IFS and proved that the k-cut set of IFS had the same properties as cut set of Zadeh’s fuzzy set. Secondly, we built up the decomposition theorem, representation theorem and extension principle of IFS. We have shown the superiority of k-cut set and representation theorem in this paper. If the theory of Zadeh’s fuzzy sets was based on the classical sets and nested sets, then it seemed to obtain a conclusion that the theory of IFS was based on IFSS and INS. Therefore, our work presented a useful approach to the theory of IFS. In future research works on IFS, one should consider to determine the membership degree lA ðxÞ and the non-membership degree mA ðxÞ of an IFS. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
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Please cite this article in press as: X.-h. Yuan et al., The theory of intuitionistic fuzzy sets based on the intuitionistic fuzzy special sets, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.02.044