<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2811508pt" id="M1" height="8.47856pt" version="1.1" viewBox="-0.0657574 -8.19741 14.2946 8.47856" width="14.2946pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg>-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets

Chen, Juanjuan; Li, Shenggang; Ma, Shengquan; Wang, Xueping

doi:https://doi.org/10.1155/2014/416530

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Volume 2014 | Article ID 416530 | https://doi.org/10.1155/2014/416530

-Polar Fuzzy Sets: An Extension of Bipolar Fuzzy Sets

Juanjuan Chen,^1,2Shenggang Li,¹Shengquan Ma,¹and Xueping Wang³

Academic Editor: Jianming Zhan

Received11 Apr 2014

Accepted23 May 2014

Published12 Jun 2014

Abstract

Recently, bipolar fuzzy sets have been studied and applied a bit enthusiastically and a bit increasingly. In this paper we prove that bipolar fuzzy sets and -sets (which have been deeply studied) are actually cryptomorphic mathematical notions. Since researches or modelings on real world problems often involve multi-agent, multi-attribute, multi-object, multi-index, multi-polar information, uncertainty, or/and limit process, we put forward (or highlight) the notion of -polar fuzzy set (actually, -set which can be seen as a generalization of bipolar fuzzy set, where is an arbitrary ordinal number) and illustrate how many concepts have been defined based on bipolar fuzzy sets and many results which are related to these concepts can be generalized to the case of -polar fuzzy sets. We also give examples to show how to apply -polar fuzzy sets in real world problems.

1. Introduction and Preliminaries

Set theory and logic systems are strongly coupled in the development of modern logic. Classical logic corresponds to the crisp set theory, and fuzzy logic is associated with fuzzy set theory which was proposed by Zadeh in his pioneer work [1].

Definition 1. An -subset (or an -set) on the set is a synonym of a mapping , where is a lattice (cf. [2]). When (the ordinary closed unit interval with the ordinary order relation), an -set on will be called a fuzzy set on (cf. [1]).
The theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, pattern recognition, robotics, computer networks, decision making, and automata theory.
An extension of fuzzy set, called bipolar fuzzy set, was given by Zhang [3] in 1994.

Definition 2 (see Zhang [3]). A bipolar fuzzy set is a pair , where and are any mappings. The set of all bipolar fuzzy sets on is denoted by BF.

Bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is . In a bipolar fuzzy set, the membership degree of an element means that the element is irrelevant to the corresponding property, the membership degree of an element indicates that the element somewhat satisfies the property, and the membership degree of an element indicates that the element somewhat satisfies the implicit counter-property. The idea which lies behind such description is connected with the existence of “bipolar information” (e.g., positive information and negative information) about the given set. Positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. Actually, a wide variety of human decision making is based on double-sided or bipolar judgmental thinking on a positive side and a negative side. For instance, cooperation and competition, friendship and hostility, common interests and conflict of interests, effect and side effect, likelihood and unlikelihood, feedforward and feedback, and so forth are often the two sides in decision and coordination. In the traditional Chinese medicine (TCM for short), “yin” and “yang” are the two sides. Yin is the feminine or negative side of a system and yang is the masculine or positive side of a system. The coexistence, equilibrium, and harmony of the two sides are considered a key for the mental and physical health of a person as well as for the stability and prosperity of a social system. Thus bipolar fuzzy sets indeed have potential impacts on many fields, including artificial intelligence, computer science, information science, cognitive science, decision science, management science, economics, neural science, quantum computing, medical science, and social science (cf. [4–45]). In recent years bipolar fuzzy sets seem to have been studied and applied a bit enthusiastically and a bit increasingly (cf. [4–45]). This is the chief motivation for us to introduce and study -polar fuzzy sets.

The first object of this note is to answer the following question on bipolar fuzzy sets.

Question 1. Is bipolar fuzzy set a very intuitive -set?
The answer to Question 1 is positive. We will prove in this note that there is a natural one-to-one correspondence between BF and (for the set of all -sets on , see Theorem 5) which preserves all involved properties. This makes the notion of bipolar fuzzy set more intuitive. Since properties of -sets have already been studied very deeply and exhaustively, this one-to-one correspondence may be beneficial for both researchers interested in above-mentioned papers and related fields (because they can use these properties directly and even cooperate with theoretical fuzzy mathematicians for a possible higher-level research) and theoretical fuzzy mathematicians as well (because cooperation with applied fuzzy mathematicians and practitioners probably makes their research more useful).
We notice that “multipolar information” (not just bipolar information which corresponds to two-valued logic) exists because data for a real world problem are sometimes from agents (). For example, the exact degree of telecommunication safety of mankind is a point in () because different person has been monitored different times. There are many other examples: truth degrees of a logic formula which are based on logic implication operators (), similarity degrees of two logic formulas which are based on logic implication operators (), ordering results of a magazine, ordering results of a university, and inclusion degrees (resp., accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision performance evaluations) of a rough set. Thus our second object of this note is to answer the following question on extensions of bipolar fuzzy sets.

Question 2. How to generalize bipolar fuzzy sets to multipolar fuzzy sets and how to generalize results on bipolar fuzzy sets to the case of multipolar fuzzy sets?
The idea to answer Question 2 is from the answer to Question 1, intuitiveness of the point-wise order on (see Remark 3), and the proven corresponding results on bipolar fuzzy sets. We put forward the notion of -polar fuzzy set (an extension of bipolar fuzzy set) and point out that many concepts which have been defined based on bipolar fuzzy sets and many results related to these concepts can be generalized to the case of -polar fuzzy sets (see Remarks 7 and 8 for details).
Apart from the backgrounds (e.g., “multipolar information”) of -polar fuzzy sets, the following question on further applications (particularly, further applications in real world problems) of -polar fuzzy sets should also be considered.

Question 3. How to find further possible applications of -polar fuzzy sets in real world problems?
Question 3 can be answered as in the case of bipolar fuzzy sets since researches or modelings on real world problems often involve multiagent, multiattribute, multiobject, multi-index, multipolar information, uncertainty, or/and limits process. We will give examples to demonstrate it (see Examples 9–14).

Remark 3. In this note (-power of ) is considered a poset with the point-wise order , where is an arbitrary ordinal number (we make an appointment that when ), (which is actually very intuitive as illustrated below) is defined by for each (), and is the th projection mapping ().(1)When , is the ordinary closed unit square in Euclidean plane . The righter (resp., the upper) a point in this square is, the larger it is. Let (the smallest element of ), , , and (the largest element of ). Then for all (especially, and hold). Notice that because both and hold. The order relation on can be illustrated in at least two ways (see Figure 1).(2)When , the order relation on can be illustrated in at least one way (see Figure 2 for the case , where , ).

2. Main Results

In this section we will prove that a bipolar fuzzy set is just a very specific -set, that is, -set. We also put forward (or highlight) the notion of -polar fuzzy set (which is still a special -set, i.e., -set, although it is a generalization of bipolar fuzzy set) and point out that many concepts which have been defined based on bipolar fuzzy sets and results related to these concepts can be generalized to the case of -polar fuzzy sets.

Definition 4. An -polar fuzzy set (or a -set) on is exactly a mapping . The set of all -polar fuzzy sets on is denoted by .

The following theorem shows that bipolar fuzzy sets and -polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one.

Theorem 5. Let be a set. For each bipolar fuzzy set on , define a -polar fuzzy set on by putting Then we obtain a one-to-one correspondence its inverse mapping is given by , , and .

Proof. Obviously, both and are mappings. For each BF, which means . Again, for each and each , which means .

Example 6. Let be a bipolar fuzzy set, where is a six-element set and and are defined by Then the corresponding -polar fuzzy set on is
In the rest of this note, we investigate the possible applications of -polar fuzzy sets. First we consider the theoretic applications of -polar fuzzy sets. More precisely, we will give some remarks to illustrate how many concepts which have been defined based on bipolar fuzzy sets and results related to these concepts can be generalized to the case of -polar fuzzy sets (see the following Remarks 7 and 8).

Remark 7. The notions of bipolar fuzzy graph (see [4, 45]) and fuzzy graph (see [46, 47]) can be generalized to the convenient (because it allows a computing in computers) and intuitive notion of -polar fuzzy graph. An -polar fuzzy graph with an underlying pair (where is symmetric; i.e., it satisfies ) is defined to be a pair , where (i.e., an -polar fuzzy set on ) and (i.e., an -polar fuzzy set on ) satisfy ; is called the -polar fuzzy vertex set of and is called the -polar fuzzy edge set of . An -polar fuzzy graph with an underlying pair and satisfying and is called a simple -polar fuzzy graph, where is the smallest element of . An -polar fuzzy graph with an underlying pair and satisfying is called a strong -polar fuzzy graph. The complement of a strong -polar fuzzy graph (which has an underlying pair ) is a strong -polar fuzzy graph with an underlying pair , where is defined by (, )
Give two -polar fuzzy graphs (with underlying pairs and , resp.) and . A homomorphism from to is a mapping which satisfies and . An isomorphism from to is a bijective mapping which satisfies and . A weak isomorphism from to is a bijective mapping which is a homomorphism and satisfies . A strong -polar fuzzy graph is called self-complementary if (i.e., there exists an isomorphism between and its complement ).
It is not difficult to verify the following conclusions (some of which generalize the corresponding results in [1, 45]).(1)In a self-complementary strong -polar fuzzy graph (with an underlying pair ), we have (2)A strong -polar fuzzy graph (with an underlying pair ) is self-complementary if and only if it satisfies (3)If and are strong -polar fuzzy graphs, then if and only if .(4)Let and be strong -polar fuzzy graphs. If there is a weak isomorphism from to , then there is a weak isomorphism from to .

Remark 8. The fuzzifications or bipolar fuzzifications of some algebraic concepts (such as group, -algebra, incline algebra (cf. [48]), ideal, filter, and finite state machine) can be generalized to the case of -polar fuzzy sets. An -polar fuzzy set is called an -polar fuzzy subgroup of a group if it satisfies . An -polar fuzzy set is called an -polar fuzzy subalgebra of a -algebra if it satisfies . An -polar fuzzy set is called an -polar fuzzy subincline of an incline if it satisfies ; it is called an -polar fuzzy ideal (resp., an -polar fuzzy filter) of if it is an -polar fuzzy subincline of and satisfies whenever (resp., satisfies whenever ). An -polar fuzzy finite state machine is a triple , where and are finite nonempty sets (called the set of states and the set of input symbols, resp.) and is any -polar fuzzy set on . Moreover, if is an -polar fuzzy set on satisfying then is called an -polar subsystem of . Furthermore, let be the set of all words of elements of of finite length and be the empty word in (cf. [28]). Then one can define a -polar fuzzy set on by putting where is the biggest element of .
The following conclusions hold.(1)An -polar fuzzy set is an -polar fuzzy subgroup of a group if and only if is or is a subgroup of .(2)An -polar fuzzy set is an -polar fuzzy subalgebra of a -algebra if and only if is or is a subalgebra of .(3)An -polar fuzzy set is an -polar fuzzy subincline (resp., an -polar fuzzy ideal, an -polar fuzzy filter) of an incline if and only if is a subincline (resp., ideal, filter) of .(4)Let be an -polar fuzzy finite state machine and be an -polar fuzzy set on . Then is an -polar subsystem of if and only if . Please see [49, 50] for more results.
Next we consider the applications of -polar fuzzy sets in real world problems.

Example 9. Let be a set consisting of five patients , , , , and (thus ). They have diagnosis data consisting of three aspects, diagnosis datum of is , where datum represents “normal” or “OK.” Suppose , , , and . Then we obtain a -polar fuzzy set which can describe the situation; this -polar fuzzy set can also be written as follows:

Example 10. -polar fuzzy sets can be used in decision making. In many decision making situations, it is necessary to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company decides which product design to manufacture, and when a democratic country elects its leaders. For instance, we consider here only the case of election. Let be the set of voters and be the set of all the four candidates. Suppose the voting is weighted. For each candidate , a voter in can send a value in to , but a voter in can only send a value in to . Suppose (which means the preference degrees of corresponding to , and are , and , resp.), , , and . Then we obtain a -polar fuzzy set which can describe the situation; this -polar fuzzy set can also be written as follows:

Example 11. -polar fuzzy sets can be used in cooperative games (cf. [51]). Let be the set of agents or players (), be the set of the grand coalitions, and be an -polar fuzzy set, where is the degree of player participating in coalition (). Again let (the set of all real numbers) be a mapping satisfying . Then the mapping is called a cooperative game, where represents the amount of money obtained by player under the coalition participating ability ().
(1) (a public good game; compare with [51, Example 6.5]) Suppose agents want to create a facility for joint use. The cost of the facility depends on the sum of the participation levels (or degrees) of the agents and it is described by where is a continuous monotonic increasing function with and is a mapping. Let be a mapping satisfying (if ) or (otherwise) . Then a cooperative game model is established, where is defined by and the function is continuously monotonic increasing with (. Obviously, the gain of agent (with participation level ) is and the total gain is
(2) There are two goods, denoted and , and three agents , , and with endowments , , and . Let be any mapping satisfying . Then the corresponding cooperative game model is , where

Example 12. -polar fuzzy sets can be used to define weighted games. A weighted game is a -tuple , where is the set of players or voters (), is a collection of fuzzy sets on (called coalitions) such that is upper set (i.e., a fuzzy set on belongs to if for some ), is an -polar fuzzy set on (called voting weights), and (called quotas). Imagine a situation: three people, , , and , vote for a proposal on releasing of a student. Suppose that casts 200 US Dollars and lose 80 hairs on her head votes each, casts 60000 US Dollars and 100 grams Cordyceps sinensis votes each, casts 100000 US Dollars and 100 grams gold votes each. Then an associated weighted game model is , where and is a collection of fuzzy sets on with an upper set, ,
(1) If the situation is a little simple, casts US Dollars (i.e., the cast is between US Dollars and US Dollars, where is an interval number which can be looked as a point ) votes each, casts US Dollars votes each, casts US Dollars votes each, and quota is . Then the corresponding weighted game model is , where is a collection of fuzzy sets on with an upper set, , and
(2) If the situation is more simple, casts US Dollars votes each, casts US Dollars votes each, casts US Dollars votes each, and quota is . Then the corresponding weighted game model is , where , , and Notice that the subset is exactly a fuzzy set on defined by and .

Example 13. -polar fuzzy sets can be used as a model for clustering or classification. Consider a set consisting of students in Chinese middle school. For a student , we use integers (resp., ) in to denote the average score of Mathematics (resp., Physics, Chemistry, Biology, Chinese, and English), and Then we obtain a -polar fuzzy set model , which can be used for clustering or classification of these students.

Example 14. -polar fuzzy sets can be used to define multivalued relations.(1)Consider a set consisting of net users (resp., patients) . For net users (resp., patients) , we use to denote the similarity between and in th aspect (), and let . Then we obtain an -polar fuzzy set , which is a multivalued similarity relation.(2)Consider a set consisting of people in a social network. For , we use to denote the degree of connection between and in th aspect (), and let . Then we obtain an -polar fuzzy set , which is a multivalued social graph (or multivalued social network) model.

3. Conclusion

In this note, we show that the enthusiastically studied notion of bipolar fuzzy set is actually a synonym of a -set (we call it -polar fuzzy set), and thus we highlight the notion of -polar fuzzy set (actually a -set, ). The -polar fuzzy sets not only have real backgrounds (e.g., “multipolar information” exists) but also have applications in both theory and real world problems (which have been illustrated by examples).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the International Science and Technology Cooperation Foundation of China (Grant no. 2012DFA11270) and the National Natural Science Foundation of China (Grant no. 11071151).

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Copyright © 2014 Juanjuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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