2.5. The Applications of Quotient Space Theory
2.5.1. Introduction
In this section, we will present a new constructional definition of fuzzy sets by using fuzzy equivalence relations, and discuss its properties. We will also discuss the isomorphism and similarity principles of fuzzy sets, the necessary and sufficient condition of the isomorphism and ε-similarity of two fuzzy equivalence relations. These principles and results can overcome the subjectivity of the concept of fuzzy sets defined by membership functions to a certain extent, and deepen the understanding of fuzzy set theory so that a more objectively fuzzy set theory can be established probably. More details can be seen in (Zhang and Zhang, 2003b, 2003d, 2005b, 2005c).
Zadeh (1965) presented a fuzzy set theory that is an efficient tool for describing uncertainty, and has wide applications, for example, fuzzy control, fuzzy reasoning, etc. But the concept of fuzziness defined by membership functions that many people pay close attention to is subjective. The definition of fuzziness that Zadeh proposed is the following.
Definition 2.21
X is a domain. If 
is a fuzzy subset of X, for any 
, assigning a number 
to x, 
is called a membership of x with respect to 
.
is a fuzzy subset of X, for any , assigning a number to x, is called a membership of x with respect to .Mapping 
, 
is called a membership function of 
.
, is called a membership function of .It is noted that in the following discussion, domain X is assumed to be infinite (not limited to finite). For simplicity, the membership function is denoted by 
rather than 
.
rather than .The main operations of fuzzy sets: given two fuzzy sets 
and 
, the union, intersection, and complement operations are defined as follows:
and , the union, intersection, and complement operations are defined as follows:
In practice such as fuzzy control, designers may choose the membership functions optionally in some degree, i.e., the membership functions of the same fuzzy variable may (slightly) be different, but the controllers designed from the different membership functions still have the same (or approximate) performances. The robustness of the fuzzy analysis method, based on (more or less) optionally chosen membership functions, has brought many people’s attention. Some researchers (Liang and Song, 1996; Lin and Tsumoto, 2000; Mitsuishi et al., 2000; Verkuilen, 2001; Lin, 2001a) presented the probabilistic interpretation of membership functions. For example, Lin (2001a) interpreted memberships as probabilities. Each sample space has a probability, and each point is associated with one sample space. So the total space is like a fiber space. Each fiber space is a probability space. Liang and Song (1996) regarded the values of a membership function as independent and identically distributed random variables, and proved that the mean of the membership function exists for all the elements of the universe of discourse. He interpreted the meaning of a subjective concept of a group of people as the mean of a membership function for all people within the group. Mitsuishi et al. (2000) introduced a new concept of empty fuzzy set, in order to define the membership functions in the probabilistic sense. That is, although different persons may assign (slightly) different membership functions to a fuzzy concept, when solving a real problem (fuzzy control or fuzzy reasoning), in average they can get an approximate result. Unfortunately, these results were obtained based on a strong assumption, i.e., the values of a membership function are assumed to be independent and identically distributed random variables. Verkuilen (2001) introduced the concept of membership functions by the multi-scale method, and discussed its corresponding properties. Lin (1988, 1992, 1997) presented a topological definition, topological rough set, of fuzzy sets by using neighborhood systems, discussed the properties of fuzzy sets from their structure, and then presented a definition of the equivalence between two fuzzy membership functions, and the necessary and sufficient conditions of the equivalence between two membership functions. He also discussed the concept of granular fuzzy sets in Lin (1998, 2001b), and ‘elastic’ membership functions in Lin (1996, 2000, 2001b). Lin’s works provide a structural interpretation of membership functions (fuzzy sets).
It can be seen that the membership function of a fuzzy set can be interpreted in two ways: one probabilistic, the other structural. We will show below that for a fuzzy set (concept), it may probably be described by different types of membership functions, as long as their structures (see the structural definition of fuzzy sets below) are the same, it still appears with the same characteristics. That is, although these membership functions are different in appearance, they are the same in essence. Therefore, the structural interpretation of fuzzy sets would be better than the probabilistic one. And it seems that in a given environment, most persons would have a similar structural interpretation for the same fuzzy concept. We will introduce a structural definition of fuzzy sets, and discuss its properties below, since the structural description is more essential to a fuzzy set.
2.5.2. The Structural Definition of Fuzzy Sets
2.5.2.1. The Structural Definition of Membership Functions
Definition 2.22
is a fuzzy equivalence relation on X. A is a crisp set of X. Define a corresponding fuzzy set A such that its membership function A(x) is as follows
(2.11)
A(x) is called a structural definition of membership functions.
Therefore, A is a fuzzy set extended from a crisp set A by fuzzy equivalence relation R and with A as its core. The new definition is induced from a fuzzy equivalence relation, and represents the relationship between a crisp set and its corresponding fuzzy set so that it deepens the understanding of fuzzy sets.
The following example shows that two different equivalence relations may correspond to the same hierarchical structure.
Example 2.16
. Given two fuzzy equivalence relations 
and 
on X as follows.Fuzzy equivalence relation 
: 
, 
, 
.
: , , .Its corresponding hierarchical structure is 
, 
, 
, and 
.
, , , and .Fuzzy equivalence relation 
is 
, 
, 
, and 
.
is , , , and .Its corresponding hierarchical structure is 
, 
, 
, and 
.
, , , and .Fuzzy equivalence relations 
and 
are different, but they have the same hierarchical structure {{1,2,3,4},{(1,2),3,4},{(1,2),(3,4)},{(1,2,3,4)}}. Certainly, there are countless fuzzy equivalence relations corresponding to the above hierarchical structure.
and are different, but they have the same hierarchical structure {{1,2,3,4},{(1,2),3,4},{(1,2),(3,4)},{(1,2,3,4)}}. Certainly, there are countless fuzzy equivalence relations corresponding to the above hierarchical structure.Definition 2.23
Definition 2.24
If 
and 
are two fuzzy equivalence relations corresponding to the same hierarchical structure, then 
and 
are called isomorphic.
and are two fuzzy equivalence relations corresponding to the same hierarchical structure, then and are called isomorphic.Definition 2.25
Given a fuzzy subset A and its membership function 
. Defining an equivalence relation 
on X, we have a quotient space 
corresponding to R. Furthermore, define an order ‘<’ on 
such that 
, space 
obtained is a totally ordered quotient space corresponding to fuzzy set A.
. Defining an equivalence relation on X, we have a quotient space corresponding to R. Furthermore, define an order ‘<’ on such that , space obtained is a totally ordered quotient space corresponding to fuzzy set A.Definition 2.26
If two fuzzy subsets 
and 
correspond to the same totally ordered quotient space, then 
and 
are called isomorphic.
and correspond to the same totally ordered quotient space, then and are called isomorphic.2.5.2.2. The Isomorphism Discrimination of Two Fuzzy Equivalence Relations
We will show below the properties of two equivalent fuzzy equivalence relations, and the necessary and sufficient condition of isomorphism for two fuzzy equivalence relations.
Proposition 2.14
If two fuzzy equivalence relations R1 and R2 on X are isomorphic, then we have the following properties.
1. For 
, 
.
, .2. 
3. For 
, 
.
, .For simplicity, in the following discussion, we use the symbol 
instead of 
.
instead of .Proof:
1. Assume 
. Let 
. f is an isomorphic mapping.
. Let . f is an isomorphic mapping.
Since 
and 
are isomorphic, there exists a one-one, strictly increasing, and onto mapping 
. Letting 
, we have x and y equivalent in 
, but u and v are not equivalent in 
. Thus, 
and 
, i.e., 
.
and are isomorphic, there exists a one-one, strictly increasing, and onto mapping . Letting , we have x and y equivalent in , but u and v are not equivalent in . Thus, and , i.e., .Similarly, we have 
.
.2. Let 
and 
. We will show 
.
and . We will show .Otherwise, in assuming that 
, for 
, let 
. Since f is a one-one, strictly increasing and onto mapping, we have 
. In assuming 
, we have 
. Thus, 
. Again, from 
, we have 
. This is a contradiction. So 
.
, for , let . Since f is a one-one, strictly increasing and onto mapping, we have . In assuming , we have . Thus, . Again, from , we have . This is a contradiction. So .Since f is a one-one mapping, we have
3. Property 3 is the deduction of Property 1. Otherwise, there exist 
and 
. But from Property 1, we have 
. This is a contradiction. So Property 3 holds.
and . But from Property 1, we have . This is a contradiction. So Property 3 holds.Definition 2.27
R1 and R2 are two fuzzy equivalence relations on X. Property 1 of Proposition 2.14 holds. Let 
and 
be the ranges of R1 and R2, respectively. Define 
as 
.
and be the ranges of R1 and R2, respectively. Define as .From Proposition 2.14, when Property 1 holds, Property 3 holds as well. From Definition 2.27, assume that 
. From Property 3, we have 
. Then, 
. Therefore, mapping f is defined as single valued and unique. From Property 1 of Proposition 2.14, f is a one-one and strictly increasing.
. From Property 3, we have . Then, . Therefore, mapping f is defined as single valued and unique. From Property 1 of Proposition 2.14, f is a one-one and strictly increasing.Proposition 2.15
R1 and R2 are two fuzzy equivalence relations on X, and satisfy Property 1 of Proposition 2.14. Let 
and 
be the ranges of R1 and R2, respectively. Then, the mapping f defined by Definition 2.27 can be extended to a one-one, strictly increasing, and onto mapping from 
.
and be the ranges of R1 and R2, respectively. Then, the mapping f defined by Definition 2.27 can be extended to a one-one, strictly increasing, and onto mapping from .Proof:
Letting 
and 
are closures of 
and 
, respectively, set 
, is open and may be represented by the union of at most countable many open intervals. Their starting and finishing points are 
and 
, respectively.
and are closures of and , respectively, set , is open and may be represented by the union of at most countable many open intervals. Their starting and finishing points are and , respectively.For any 
, there exists 
. Define 
.
, there exists . Define .We will show next that 
must be some 
.
must be some .Otherwise, in assuming that 
, there exists 
(monotonously decrease to 
). From Property 1 and 2, we have 
(monotonously decrease to 
). Then there exists 
, when 
, 
.
, there exists (monotonously decrease to ). From Property 1 and 2, we have (monotonously decrease to ). Then there exists , when , .This contradict with 
. Then, we have that 
is some 
.
. Then, we have that is some .Similarly, when 
, we have 
.
, we have .Now, f will be expanded to [0,1]. For 
, let 
. When 
let 
. And let 
. When 
assume that 
. Letting 
we have 
. When 
define 
.
, let . When let . And let . When assume that . Letting we have . When define .Therefore, f is expanded to a one-one, strictly increasing, and onto mapping from 
.
.Assume that 
and 
are hierarchical structures corresponding to 
and 
, respectively, then 
.
and are hierarchical structures corresponding to and , respectively, then .Theorem 2.7: The Discrimination Principle of Isomorphism
R1 and R2 are two fuzzy equivalence relations on X. The necessary and sufficient conditions of isomorphism of R1 and R2 are that the following two properties hold.
1. Property 1 of Proposition 2.14
2. Property 2 of Proposition 2.14
Proof:
: Let f be the mapping defined in Definition 2.27. From Proposition 2.15, it’s known that f can be expanded to [0,1], and is still a one-one, strictly increasing, and onto mapping. Thus, f is an isomorphic mapping of R1 and R2.The expansion process is the following.
From Property 2 of Proposition 2.14, for point 
, let 
. Letting 
, we define 
. For 
, i.e., 
is an external point of S, there exists an interval 
such that 
.
, let . Letting , we define . For , i.e., is an external point of S, there exists an interval such that .Letting 
and 
, we have
and , we have
Letting 
, and 
, define 
.
, and , define .Therefore, f is expanded to a one-one, strictly increasing, and onto mapping from 
, i.e., an isomorphic mapping corresponding to R1 and R2.
, i.e., an isomorphic mapping corresponding to R1 and R2.
: From Proposition 2.14, Property 1 and 2 hold.Proposition 2.16
The ranges of R1 and R2 are [0,1]. The isomorphism of R1 and R2 ⇔ their corresponding function 
is a strictly increasing and continuous function.
is a strictly increasing and continuous function.Proof:
⇐: Since f is a strictly increasing and continuous function, for any 
, and Property 2 of Proposition 2.14 holds. From Proposition 2.15, R1 and R2 are isomorphic.
, and Property 2 of Proposition 2.14 holds. From Proposition 2.15, R1 and R2 are isomorphic.⇒: Assume that R1 and R2 are isomorphic. From the Basic Theorem, it’s known that their corresponding isomorphic mapping 
is a one-one, strictly increasing, and onto mapping. So f is continuous.
is a one-one, strictly increasing, and onto mapping. So f is continuous.Corollary 2.2
Assume 
is a distance function on 
. Two fuzzy equivalence relations on X are defined as follows.
are strictly decreasing and continuous functions. Then, R1 and R2 are isomorphic.
is a distance function on . Two fuzzy equivalence relations on X are defined as follows.
are strictly decreasing and continuous functions. Then, R1 and R2 are isomorphic.Proof:
We have 
, where 
, and 
is the inverse of 
. Since 
and 
are strictly decreasing and continuous, 
is also strictly decreasing and continuous. Since 
is the combination of two strictly decreasing and continuous functions, 
is strictly increasing and continuous. Then, we have that R1 and R2 are isomorphic.
, where , and is the inverse of . Since and are strictly decreasing and continuous, is also strictly decreasing and continuous. Since is the combination of two strictly decreasing and continuous functions, is strictly increasing and continuous. Then, we have that R1 and R2 are isomorphic.Corollary 2.3
Assume 
is a distance function on 
. Two fuzzy equivalence relations on X are defined as follows.
.
is a distance function on . Two fuzzy equivalence relations on X are defined as follows.
.If 
, then R1 and R2 are isomorphic.
, then R1 and R2 are isomorphic.Example 2.17
Assume 
. Let
. Let
Choose a proper 
such that 
.
such that .
We have 
, 
, 
, and 
. From Corollary 2.3, R1 and R2 are isomorphic.
, , , and . From Corollary 2.3, R1 and R2 are isomorphic.2.5.3. The Robustness of the Structural Definition of Fuzzy Sets
2.5.3.1. The Isomorphism of Fuzzy Sets
Proposition 2.17
R1 and R2 are two isomorphic fuzzy equivalence relations on X. f is isomorphic transformation. A is a common set on X. For 
, fuzzy subsets 
and 
are defined by 
and 
, according to Definition 2.22. Then, 
and 
are isomorphic, and 
.
, fuzzy subsets and are defined by and , according to Definition 2.22. Then, and are isomorphic, and .Proof:
From Definition 2.22, we have
From 
and that f is one-one and strictly increasing, there exists 
such that
and that f is one-one and strictly increasing, there exists such that
Again, sine f is strictly increasing and continuous, we have 
. Thus, 
and 
.
. Thus, and .Similarly, we have 
, i.e., 
, i.e., Finally, 
.
.Proposition 2.18
R1 and R2 are two isomorphic fuzzy equivalence relations on X. A and B are two common sets on X. Fuzzy subsets 
and 
are defined by 
and 
, according to Definition 2.22. Then, 
and 
(or 
and 
) are isomorphic.
and are defined by and , according to Definition 2.22. Then, and (or and ) are isomorphic.Proof:
and 
are membership functions corresponding to the four fuzzy subsets. The membership functions of 
and 
are denoted by 
and 
, respectively. From Definition 2.22, we have 
and 
.In order to show that 
and 
are isomorphic, it is only needed to show 
, 
. Since f is one-one and strictly increasing, from Proposition 2.15, we have
and are isomorphic, it is only needed to show , . Since f is one-one and strictly increasing, from Proposition 2.15, we have
Therefore, 
and 
are isomorphic.
and are isomorphic.Similarly, 
and 
are isomorphic.
and are isomorphic.Proposition 2.19
R1 and R2 are two isomorphic fuzzy equivalence relations on X. A and B are two common sets on X. Fuzzy subsets 
and 
are defined by 
and 
, according to Definition 2.22, then 
and 
are isomorphic, where 
is the complement of A1, and its membership function is 
.
and are defined by and , according to Definition 2.22, then and are isomorphic, where is the complement of A1, and its membership function is .Proof:
Since the corresponding totally ordered quotient space of the complement of 
has the same elements with that of 
and the only difference is their opposite order, 
and 
are isomorphic.
has the same elements with that of and the only difference is their opposite order, and are isomorphic.Theorem 2.8: Isomorphism Principle
R1 and R2 are two isomorphic fuzzy equivalence relations, and 
is a set of common subsets on X. Define two sets 
and 
of fuzzy subsets from 
and 
respectively, and carry out a finite number of union, intersection and complement operations on them, we have two sets 
and 
of fuzzy subsets. Then, the two sets C and D of fuzzy sets are isomorphic.
is a set of common subsets on X. Define two sets and of fuzzy subsets from and respectively, and carry out a finite number of union, intersection and complement operations on them, we have two sets and of fuzzy subsets. Then, the two sets C and D of fuzzy sets are isomorphic.Proof:
The conclusion can be obtained from Propositions 2.15 and 2.16.
The theorem shows that the corresponding totally ordered quotient space of a fuzzy subset is its essential property. For the isomorphic fuzzy subsets, although they might have different membership functions, after a finite number of operations (inference), the fuzzy subsets obtained are still isomorphic. Therefore, as long as the essential property of fuzzy subsets remains the same, the usage of different membership functions does not affect the final results.
Example 2.18
Define fuzzy subsets 
and 
from 
and 
, respectively. Their membership functions as follows.
and from and , respectively. Their membership functions as follows.
Then we have
Fuzzy sets 
and 
have different membership functions on X, but they are isomorphic since they correspond to the same totally ordered quotient space 
.
and have different membership functions on X, but they are isomorphic since they correspond to the same totally ordered quotient space .Similarly, fuzzy sets 
and 
also have different membership functions, but they correspond to the same totally ordered quotient space 
as well.
and also have different membership functions, but they correspond to the same totally ordered quotient space as well.From the structural relation view point, both fuzzy subsets 
and 
indicate that elements 1 and 3 belong to their core A, and element 2 is closer to the core than element 4. Due to the different understanding of each person, their descriptions of the degree of closeness might be different, but the mutual relationships among elements should be the same.
and indicate that elements 1 and 3 belong to their core A, and element 2 is closer to the core than element 4. Due to the different understanding of each person, their descriptions of the degree of closeness might be different, but the mutual relationships among elements should be the same.The structural definition, given in Definition 2.22, of fuzzy sets is defined from fuzzy equivalence relations. This is very significant. For example, if using neighborhood systems to defined fuzzy sets directly, the isomorphism principle might not hold. The following is a counter-example.
Example 2.19
Assume that the structures of fuzzy subsets 
and 
are the following.
and are the following.
Define their membership functions as follows
, 
, 
, and 
, where 
and 
are membership functions of 
, 
and 
are membership functions of 
.Carry out set operations on 
and 
with membership functions 
, 
and 
, 
, respectively, we have
and with membership functions , and , , respectively, we have
After the set operations, the results with respect to different membership functions are no long isomorphic. This shows that using fuzzy membership functions to define fuzzy sets, although it is a structural definition but cannot satisfy the isomorphism principle generally (Lin, 1997).
2.5.3.2. The 
-Similarity of Fuzzy Sets
The Definition of 
-Similarity of Fuzzy Sets
In the above section, we presented a structural definition of fuzzy sets, which answers the cause of the robustness of fuzzy analysis in some degree. But the conditions that two fuzzy equivalence relations are isomorphic are too strong. We present a weaker condition, i.e., 
-similarity of fuzzy sets, below.
-similarity of fuzzy sets, below.Definition 2.28
Given two fuzzy equivalence relations 
and 
, and 
. If there exists fuzzy equivalence relation 
such that (1) 
and 
are isomorphic, (2) 
, then 
and 
are called 
-similarity.
and , and . If there exists fuzzy equivalence relation such that (1) and are isomorphic, (2) , then and are called -similarity.Definition 2.29
Given two fuzzy equivalence relations 
and 
, and 
. If there exists fuzzy set 
such that (1) 
and 
are isomorphic (or 
and 
are isomorphic), (2) 
(or 
), then 
and 
are called 
-similarity.
and , and . If there exists fuzzy set such that (1) and are isomorphic (or and are isomorphic), (2) (or ), then and are called -similarity.Proposition 2.20
Assume that 
and 
are two 
-similarity fuzzy equivalence relations on X. A is a common set on X. Fuzzy subsets 
and 
are defined by 
and 
, respectively, according to Definition 2.22, then 
and 
are called 
-similarity.
and are two -similarity fuzzy equivalence relations on X. A is a common set on X. Fuzzy subsets and are defined by and , respectively, according to Definition 2.22, then and are called -similarity.Proof:
Assume that 
, i.e., 
and 
are isomorphic, and 
.
, i.e., and are isomorphic, and .For x, assume that 
. Thus
. Thus
Similarly, 
.
.That is, 
. We have that 
and 
are 
-similarity.
. We have that and are -similarity.Proposition 2.21
Assume that 
and 
are two 
-similarity fuzzy equivalence relations on X. A and B are two common sets on X. Fuzzy subsets 
, 
and 
, 
are defined from 
and 
, according to Definition 2.22. Then, 
and 
(or 
and 
) are 
-similarity.
and are two -similarity fuzzy equivalence relations on X. A and B are two common sets on X. Fuzzy subsets , and , are defined from and , according to Definition 2.22. Then, and (or and ) are -similarity.Proof:
The membership functions corresponding to the four fuzzy sets are 
and 
, respectively. The membership functions of 
and 
are denoted by 
and 
, respectively. From Definition 2.22,
and , respectively. The membership functions of and are denoted by and , respectively. From Definition 2.22,
Assume that 
, i.e., 
and 
are isomorphic, and 
.
, i.e., and are isomorphic, and .Assume that 
.
.Similarly, 
. Finally, we have 
, i.e., 
and 
are 
-similarity.
. Finally, we have , i.e., and are -similarity.Similarly, 
and 
are 
-similarity.
and are -similarity.Theorem 2.9: Similarity Principle
Proof:
Assume that 
and 
are isomorphic, and 
. Let 
be a set of fuzzy sets defined by 
. After a finite number of set operations, we have a set 
of fuzzy sets. From the isomorphism principle, C and F are isomorphic.
and are isomorphic, and . Let be a set of fuzzy sets defined by . After a finite number of set operations, we have a set of fuzzy sets. From the isomorphism principle, C and F are isomorphic.On the other hand, using the same method of Proposition 2.21, we have 
, i.e., D and C are 
-similarity, where 
and 
are membership functions of fuzzy sets 
and 
.
, i.e., D and C are -similarity, where and are membership functions of fuzzy sets and .The Discrimination of 
-Similarity of Fuzzy Sets
Theorem 2.10
Assume that the ranges of 
and 
are [0,1]. Then, we have 
and 
are 
-similarity ⇔ there exists a strictly increasing function F such that 
.
and are [0,1]. Then, we have and are -similarity ⇔ there exists a strictly increasing function F such that .Proof:
Let 
. From Proposition 2.16, we have 
and 
are isomorphic. Then, from Definition 2.28, we have 
and 
are 
-similarity.
. From Proposition 2.16, we have and are isomorphic. Then, from Definition 2.28, we have and are -similarity.3. The Structural Property of 
-Similarity of Fuzzy Equivalence Relations
Assume that 
and 
are two equivalence relations. 
and 
are their corresponding hierarchical structures. If 
and 
are 
-similarity, then there exists a strictly increasing function 
for 
such that 
.
and are two equivalence relations. and are their corresponding hierarchical structures. If and are -similarity, then there exists a strictly increasing function for such that .Conversely, for 
such that 
.
such that .
-Similarity between Two Equivalence Relations
Fig. 2.3 shows that the hierarchical structures corresponding to 
and 
cannot be merged into one structure, but for any quotient space 
within 
, there exist two quotient spaces 
and 
in 
, one is in front of 
, and the other is behind 
. Conversely, for any 
within 
, there exist two quotient spaces 
and 
in 
, one is in front of 
, and the other is behind 
.
and cannot be merged into one structure, but for any quotient space within , there exist two quotient spaces and in , one is in front of , and the other is behind . Conversely, for any within , there exist two quotient spaces and in , one is in front of , and the other is behind .2.5.3.3. The Geometrical Meaning of the Structural Definition of Fuzzy Sets
In the structural definition of fuzzy sets, their membership functions are induced from equivalence relations. Now, we discuss the geometrical meaning of structures of fuzzy sets by using structures of fuzzy equivalence relations.
A fuzzy equivalence relation 
is given. First, assume that fuzzy subset 
is induced from a singleton 
. The membership function of the fuzzy set defined by 
is 
. From Basic Theorem, letting 
, then 
is a normalized isosceles distance of some quotient space [X] of X. Under the distance, 
is the neighborhood system of 
, where 
corresponds to the structure of fuzzy set 
.
is given. First, assume that fuzzy subset is induced from a singleton . The membership function of the fuzzy set defined by is . From Basic Theorem, letting , then is a normalized isosceles distance of some quotient space [X] of X. Under the distance, is the neighborhood system of , where corresponds to the structure of fuzzy set .According to Definition 2.25, a totally ordered quotient space 
can be defined by the neighborhood system. The order of the quotient space is decided by the distance from point 
.
can be defined by the neighborhood system. The order of the quotient space is decided by the distance from point .Generally, for any set A, define a fuzzy set A based on Definition 2.22, and A(x) is its membership function. Letting 
, 
is the distance from point x to set A in the sense of distance 
. If 
is a ε-neighborhood of A, then a corresponding totally ordered quotient space can be defined by the neighborhood system 
of A, according to Definition 2.25, where the order of the quotient space is decided by the distance from a point to set A.
, is the distance from point x to set A in the sense of distance . If is a ε-neighborhood of A, then a corresponding totally ordered quotient space can be defined by the neighborhood system of A, according to Definition 2.25, where the order of the quotient space is decided by the distance from a point to set A.From the isomorphism principle in Section 2.5.2, it’s known that the structure of the totally ordered quotient space, corresponding to a fuzzy subset, is the essential property of the fuzzy subset.
Fig. 2.4 shows the geometrical intuition of common set A, and its corresponding fuzzy set A, where a crisp set A corresponds to a ‘platform’ with a vertical boundary face, while a fuzzy set A corresponds to a ‘platform’ with a slant boundary face.
From Definition 2.25, it’s known that a fuzzy set A decides an order relation of some quotient space [X] on X. The order is the essential property of A, and provides the relationship of distances from a spatial point to the core 
of fuzzy set A. In common membership functions, an absolute value is used to describe the relation between a fuzzy set and any element within the set. While in the structural definition of fuzzy sets, the relative order relationship among elements is used to describe the fuzziness. Using the structural definition, as long as people have the same understanding of the relative order relation (or 
-similarity) among elements, even different membership functions are used, the results they have are isomorphic (
-similarity), and after a finite number of fuzzy set operations, the results are still isomorphic. We show an example below.
of fuzzy set A. In common membership functions, an absolute value is used to describe the relation between a fuzzy set and any element within the set. While in the structural definition of fuzzy sets, the relative order relationship among elements is used to describe the fuzziness. Using the structural definition, as long as people have the same understanding of the relative order relation (or -similarity) among elements, even different membership functions are used, the results they have are isomorphic (-similarity), and after a finite number of fuzzy set operations, the results are still isomorphic. We show an example below.Example 2.20
Fuzzy equivalence relation 
is as Example 2.16. Letting 
, define A and its corresponding membership function as
is as Example 2.16. Letting , define A and its corresponding membership function as
Its corresponding totally ordered quotient space is {(1),(1,2),(1,2,3,4)}.
Letting 
be 
, then its corresponding totally ordered quotient space is {(1),(1,2),(1,2,3,4)}.
be , then its corresponding totally ordered quotient space is {(1),(1,2),(1,2,3,4)}.There are three elements (1),(2),(3,4) in quotient space [X]A. Their order relation is (1)<(2)<(3,4). And the order relation is the essential property of fuzzy set 
.
.2.5.4. Conclusions
In the section, from the quotient space theory, we present the structural definition of a fuzzy set based on fuzzy equivalence relations, and the isomorphism and similarity principles of fuzzy sets. These principles may interpret the cause of robustness of fuzzy analysis, and answer the question why the same (or similar) results can be had, even using different membership functions in real fuzzy analysis.
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