Introduction to Fuzzy Sets

The subject of this chapter is fuzzy sets and the basic issues related to them. The first section discusses concepts of sets: classic and fuzzy, and presents various ways of describing fuzzy sets. The second section is dedicated to t-norms, s-norms, and other terms associated with fuzzy sets. Subsequent sections describe the extension principle, fuzzy relations and their compositions, cylindrical extension and projection of a fuzzy set. The sixth section discusses fuzzy numbers and basic arithmetic operations on them. Finally, the last section summarizes the chapter.

There are several operations defined on classic sets and the following are considered to be basic ones [19]: • product (intersection, conjunction)

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• negation (complement) The above operations can also be defined on the basis of characteristic functions [19]: 1 Introduction to Fuzzy Sets • Gaussian membership function (where c and δ are parameters) (1.8) The parameter c specifies the center of a function; the parameter δ determines its dispersion. • Trapezoidal membership function (where p ≤ q ≤ r ≤ s are parameters) (1.9) A special case of a trapezoidal function (for q = r ) is a triangular function. • Singleton (where x 0 is a parameter) (1.10) The parameter x 0 specifies the location of the singleton, that is, the single value of x which belongs to a set A (with a membership degree equal to 1).
An example of the Gaussian membership function is presented in Fig. 1.1b, and trapezoidal, triangular, and singleton functions are illustrated in Fig. 1  Example 1.3 One more example concerning fetal heart rate can be the expression "normal FHR," which means that the FHR value is in the physiological range. Based on FIGO guidelines, as the range of "normal FHR" values we can assume [110,150] bpm and use the classic set of values in this range to describe the expression "normal FHR." However, this leads to the situation in which FHR value 151 bpm is not "normal," although it seems that it partially is. It suggests that it is better to use a fuzzy set to describe the expression "normal FHR." Example 1.4 Another example can be the expression "new car." Assuming that a car is "new" when its age does not exceed three years, the expression "new car" can be described by the classic set of cars up to the age of three years. However, it results in a problem similar to the previous example: the car at the age of three years and one week is not "new," although it seems that it almost is. Also in this case it is better to use a fuzzy set.
The above examples suggest that fuzzy sets are a good tool for a formal description of vague and imprecise expressions such as "value about 120," "normal FHR," "new car," "medium height," "high salary," and so on. Examples of membership functions shown in Fig. 1.2 could be used to describe expressions such as: (a) "normal FHR," (b) the value of FHR is "about 120 bpm," and (c) FHR value is "exactly 120 bpm." Another way of describing a fuzzy set is to list ordered pairs: an object x and its membership degree (1.11) To describe a fuzzy set, the notation proposed by Zadeh [25] can also be used: • for discrete universe X (comprising ordered or nonordered objects) (1.13) In the above notation the symbol / is a separator, and symbols and denote idempotent summation. Example 1.5 In the discrete nonordered universe comprising selected fruits X = {orange, pineapple, grape, apple, peach, banana, grape f ruit} let us define the fuzzy set A "Fruits, that the first author likes." Using the notation proposed by Zadeh we can write A = 1.0/orange + 0.6/ pineapple + 0.2/grape + 1.0/apple + 0.8/ peach +0.6/banana + 0.5/grape f ruit. Example 1.6 Let us consider the discrete ordered universe comprising values of possible temperatures to set in a car air-conditioning system X = {low, 18, 19, . . . , 23 24, high} ⊂ R + , where "low" and "high" mean the lowest and the highest attainable temperatures. Using the notation of ordered pairs, the fuzzy set A "Adequate (according to the second author) temperature in the car" defined in the universe X can be described as follows Example 1.7 The set of FHR values from Example 1.2 using the notation proposed by Zadeh is described as Various extensions of fuzzy sets were proposed, for example, fuzzy sets of type-2 [26], interval-valued fuzzy sets [8,11,21,26], probabilistic sets [9], rough sets [18], and intuitionistic fuzzy sets [2].

Fuzzy Sets-Basic Definitions
Similarly to classic sets, operations of product, sum, and complement are also established for fuzzy sets. Product and sum are defined by means of operators of t-norm and s-norm: (1.14) There are various t-norms and s-norms [5,7,13,16,23,24,27], three that are frequently used are presented below.
• Zadeh t-norm and s-norm: (1.16) • Algebraic product and algebraic (also called probabilistic) sum: • Lukasiewicz t-norm and s-norm: The complement of a fuzzy set A is defined as follows [5,16] where n denotes a negation function. Minimal assumptions about the function n are: n is a mapping [0, 1] → [0, 1], n satisfies conditions n(0) = 1, n(1) = 0, and n is  In addition to definitions of t-norms and s-norms, other concepts that also characterize fuzzy sets are defined [3, 5-7, 16, 20, 23, 27]. Some of them are discussed below.
• Support of a fuzzy set (Supp( A)), that is, the strong α-cut set with α = 0.
• Width of a fuzzy set, Width(A) = |x 2 − x 1 |, where x 1 and x 2 are crossover points of A defined below. • Crossover points of a fuzzy set (1.21) • A fuzzy set is "normal" if its core is not empty.
• Two fuzzy sets A and B are equal iff The illustration of core, support, width, and crossover points of a fuzzy set

Extension Principle
The extension principle [26] allows for extension of the concept of mathematical functions (defined on classic sets) to fuzzy sets. In other words, mapping from a classic set to a classic set is extended to mapping from a fuzzy set to a fuzzy set.
Let y = f (x) be a one-argument function, which is mapping from X to Y, and A be a fuzzy set defined in a discrete universe As a result of mapping a set A by the function f we obtain the following fuzzy set B (defined in the universe Y) [16] where + denotes a logical sum. For functions that are not injective (are not one-to-one mappings), a logical sum is performed using s-norm, and hence we can write [16] where S stands for a multiargument s-norm, and f −1 (y) denotes the domain of function y = f (x). 4 , which is the mapping from X to Y. Let us determine the mapping of A by the function f to the fuzzy set B.
Values of the function f arranged in ascending order are Using (1.27) and Zadeh s-norm (maximum), the fuzzy set B is determined in the following way The above description concerned a one-argument function. Now let us consider a general case, a multiargument function y = f (x 1 , x 2 , . . . , x n ), which is a mapping from X 1 × X 2 × · · · × X n to Y. Suppose we have fuzzy sets A 1 , A 2 , . . . , A n , defined in universes X 1 , X 2 , . . . , X n , respectively. The mapping of these sets by the function f leads to the following fuzzy set B (defined in universe Y) [16] (1.28)

Fuzzy Relations
A generalization of the concept of a fuzzy set is the idea of a fuzzy relation [26]. Let us consider a two-dimensional (binary) fuzzy relation R, which can be described by the set of ordered pairs: two objects x and y, and the membership degree μ R (x, y). It can be written as [5,16] The membership degree μ R (x, y) can be understood as the degree of relationship between objects x and y; the higher the value of μ R (x, y), the greater is the degree of relationship. The membership degree μ R (x, y) is the value of membership function μ R : X × Y → [0, 1] of fuzzy relation R. The presented two-dimensional fuzzy relation is also a two-dimensional fuzzy set defined in an universe X × Y.

Example 1.9
In universes X = Y = [100, 160] ⊂ R + let us define the twodimensional fuzzy relation R "Two FHR values (x and y) differ significantly." As the membership function of such a relation we can assume which for parameter δ = 0.2 is shown in Fig. 1.5.
Example 1.10 Let us consider two discrete universes: X = {3500, 4000, 4500, 5000} ⊂ R + and Y = {2000, 3000, 3500, 5000, 5500} ⊂ R + , comprising possible salaries in companies A and B, respectively. In universe X × Y we can define the following fuzzy relation "The salary of an employee x in a company A is similar to the salary of an employee y in a company B." A relation defined in discrete universes can be described by a relation matrix. In this case it is as follows where rows correspond to elements of the universe X, and columns to elements of the universe Y. In other words, the element R(i, j) determines the degree of relationship between the ith object in X and the jth object in Y.
In general, a multidimensional fuzzy relation is defined as follows [5,16] where μ R : X 1 × X 2 × · · · × X n → [0, 1] is a membership function of an n-dimensional fuzzy relation R, that is, of an n-dimensional fuzzy set defined in universe X 1 × X 2 × · · · × X n .
Because fuzzy relations are fuzzy sets, they are subject to the same operations as fuzzy sets, for example, the product of sets or an α-cut set. Additionally, fuzzy relations may be composed. Let R 1 and R 2 be relations defined in universes X × Y and Y × Z, respectively. Frequently used compositions are "supremum-t-norm" (R 1 • R 2 ) and "infimum-s-norm" (R 1 • R 2 ), leading to the relation defined in an universe X × Z [16]: (1.32) For relations described by relation matrices, the above compositions can be achieved by multiplication of matrices with multiplication of elements replaced by t-norm (or s-norm), and the adding of elements replaced by maximum (or minimum). The composition "maximum-t-norm" (R 1 • R 2 ) with Zadeh t-norm (minimum) leads to the following relation

Cylindrical Extension and Projection of a Fuzzy Set
When analyzing fuzzy sets (fuzzy relations) defined in universes of different dimensionality, sometimes there is a need to increase or reduce dimensionality of one of the sets (one of the relations). To increase or reduce dimensionality, operations of cylindrical extension and projection were defined [26]. Cylindrical extension of a fuzzy set A leads to a fuzzy set (denoted by Ce( A)) of higher dimensionality. Let us assume we have a fuzzy set A defined in a onedimensional universe X. Its cylindrical extension in two-dimensional universe X × Y is defined as [16] ∀ x∈X,y∈Y and is illustrated in Fig. 1.6a, which shows the cylindrical extension of the fuzzy set A from Example 1.2 in the universe X × Y, where Y = [0, 50] ⊂ R. In general, let us assume we have a fuzzy set A defined in an m-dimensional universe X = X 1 × X 2 × · · · × X m . The cylindrical extension of A in an m + ndimensional universe XY = X × Y, where Y = Y 1 × Y 2 × · · · × Y n , is defined as [16] where x XY and x X denote objects from universes XY and X, respectively. Projection of a fuzzy set [26] leads to fuzzy sets of lower dimensionality. For example, let us consider a fuzzy set A defined in a two-dimensional universe X × Y and described by the membership function presented in Fig. 1.6b. As a result of its projection in universes X and Y we can obtain two fuzzy sets [16]: In general, let us assume we have a fuzzy set A defined in an (m + n)-dimensional universe XY = X × Y. Its projection in an m-dimensional universe X is defined as [16] where x X , x Y , and x XY denote objects from universes X, Y, and XY, respectively.

Fuzzy Numbers
A separate class of fuzzy sets for describing imprecise expressions related to numbers (such as "about 5," "more or less 10," etc.) is distinguished [26]. Such sets are called fuzzy numbers and denoted by A, B, . . . [16]. Usually, fuzzy numbers are regarded as fuzzy sets that are defined over the real axis and fulfill given conditions; for example, they are normal, compactly supported, and in some sense convex [15]. Basic operations on fuzzy numbers A and B can be defined based on the extension principle in the following way [16]: (1.41) Example 1.14 Let us calculate addition, subtraction, multiplication, and division of the following fuzzy numbers.
It can be noticed that the first number represents a value "about −1" and the second one "about 5," because membership degrees for −1 and 5 are equal to 1. Useful calculations are presented in Table 1  It can be noted that the obtained results represent values: "about 4" (for the sum), "about 6" (subtraction), "about −5" (multiplication and division), which is consistent with classic arithmetic, for example, "about −1" + "about 5" = "about 4." The considered arithmetic operations were defined based on the extension principle. Alternatively α-cuts of fuzzy numbers can be used. Figure 1.7a shows an α-cut of a fuzzy set A (see (1.20) in Sect. 1.2). According to the figure, as a result of an α-cut a classic set described by the interval a − , a + is obtained. Arithmetic operations on fuzzy numbers A and B using α-cuts consist in application of interval arithmetic to intervals describing α-cuts of these numbers: A α = a − , a + and B α = b − , b + . According to [1] arithmetic operations are defined as follows:

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which are presented in Fig. 1.7b. Let us calculate addition, subtraction, multiplication, and division of A and B using their α-cuts.
which is shown in Fig. 1.7e. The result of the division is calculated similarly applying (1.45), however, there is no need to select solutions of equations since each of them has a single solution. Finally, we get the membership function which is presented in Fig. 1.7f. Analyzing membership functions of the considered fuzzy numbers A and B it can be noted that they represent values "about 3" and "about 5," because membership degrees for 3 and 5 are equal to 1. The obtained results of arithmetic operations are correct; for example, the subtraction provided value "about −2." As opposed to classic arithmetic, where two numbers are equal or are not equal, in fuzzy arithmetic a "partial equality" is possible. One of the methods of determining the degree of equality is based on the distance between compared fuzzy sets [16]. According to it, the equality index of sets A and B is defined as Eq 1 (A, B) = 1 − d p (A, B), where d p (A, B) denotes Minkowski distance between sets described by membership functions μ A (x) and μ B (x) Minkowski distance between sets is also the basis of one of the methods of ranking fuzzy numbers [16]. According to it, to compare fuzzy numbers A and B, the fuzzy number C such as A ≤ C and B ≤ C is established. The comparison of A and B consists in the analysis of their Minkowski distances from C; it is stated that A ≤ B if d p A, C ≥ d p B, C . Most often C = max A, B is established based on the extension principle [16] Another way of ranking fuzzy numbers is to use their α-cuts [23]. The extension of the concept of fuzzy numbers are Ordered Fuzzy Numbers (OFNs) proposed in [14,15]. The OFNs are ordered pairs of continuous real functions defined on the interval [0, 1] and their applications are the subject of research [4,12,17].

Summary
The chapter provides the review of basic issues concerning fuzzy sets, which -in contrast to classic sets -allow for partial membership of objects. As a result fuzzy sets are a good tool for representing vague and imprecise expressions of natural language. Various ways of describing fuzzy sets and concepts related to them were shown. We discussed the extension principle, which allows for extension of traditional mathematical functions to fuzzy sets, as well as the idea of fuzzy relation, which makes possible a formal description of the relationship between two or more fuzzy sets. Operations of cylindrical extension and projection of a fuzzy set, which enable increasing and reducing its dimensionality, were also described. A separate section was dedicated to fuzzy numbers and basic arithmetic operations on them. The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.