Lecture 3

1. Lecture 3 Fuzzy sets

2. 1.1 Sets • 1.1.1 Elements of sets • An universal set X is defined in the universe of discourse and it includes all possible elements related with the given problem. If we define a set A in the universal set X, we see the following relationships • A ⊆ X • In this case, we say a set A is included in the universal set X. If A is not included in X, this relationship is represented as follows. • If an element x is included in the set A, this element is called as a member of the set and the following notation is used. • x ∈ A • If the element x is not included in the set A, we use the following notation. • x ∉ A • In general, we represent a set by enumerating its elements. For example, elements a1, a2, a3,… , an are the elements of set A, it is represented as follows. • A = {a1, a2,… , an }

3. Another representing method of sets is given by specifying the conditions of elements. For example, if the elements of set B should satisfy the conditions P1, P2,… , Pn, then the set B is defined by the following. B = {b | b satisfies p1, p 2,… , pn } In this case the symbol .|. implies the meaning of .such that.. In order to represent the size of N-dimension Euclidean set, the number of elements is used and this number is called cardinality. The cardinality of set A is denoted by |A|. If the cardinality |A| is a finite number, the set A is a finite set. If |A| is infinite, A is an infinite set. In general, all the points in N-dimensional Euclidean vector space are the elements of the universal set X.

4. 1.1.2 Relation between sets • A set consists of sets is called a family of sets. For example, a family set containing sets A1, A2,… is represented by • {Ai | i ∈ I} • where i is a set identifier and I is an identification set. If all the elements in set A are also elements of set B, A is a subset of B. • A ⊆ B iff (if and only if) x ∈ A ⇒ x ∈ B • The symbol ⇒ means .implication.. If the following relation is satisfied, A ⊆ B and B ⊆ A A and B have the same elements and thus they are the same sets. This relation is denoted by A = B If the following relations are satisfied between two sets A and B, A ⊆ B and A ≠ B then B has elements which is not involved in A. In this case, A is called a proper subset of B and this relation is denoted by A ⊂ B A set that has no element is called an empty set ∅. An empty set can be a subset of any set.

5. 1.1.3 Membership • If we use membership function (characteristic function or discrimination function), we can represent whether an element x is involved in a set A or not. • Definition (Membership function) For a set A, we define a membership function µA such as • µA(x)= 1 if and only if x ∈ A • 0 if and only if x ∉ A • We can say that the function µA maps the elements in the universal set X to the set {0,1}. • µA : X → {0,1} □ • As we know, the number of elements in a set A is denoted by the cardinality |A|. A power set P(A) is a family set containing the subsets of set A. Therefore the number of elements in the power set P(A) is represented by • |P(A)| = 2|A| • Example 1.1 If A = {a, b, c}, then |A| = 3 • P(A) = {∅, {a}, {b}, {a, b}, {a, c}, {b, c}, {a, b, c}} • |P(A)| =23 = 8 □

6. 1.2 Operation of Sets • 1.2.1 Complement • The relative complement set of set A to set B consists of the elements which are in B but not in A. The complement set can be defined by the following formula. • If the set B is the universal set X, then this kind of complement is an absolute complement set . That is, • In general, a complement set means the absolute complement set. The complement set is always involutive. • The complement of an empty set is the universal set, and vice versa.

7. 1.2.2 Union • The union of sets A and B is defined by the collection of whole elements of A and B. • A ∪ B = {x | x ∈ A or x ∈ B} • The union might be defined among multiple sets. For example, the union of the sets in the following family can be defined as follows. • where the family of sets is {Ai | i ∈ I} • The union of certain set A and universal set X is reduced to the universal set. • A ∪ X = X • The union of certain set A and empty set ∅ is A. • A ∪ ∅ = A • The union of set A and its complement set is the universal set.

8. 1.2.3 Intersection • The intersection A ∩ B consists of whose elements are commonly included in both sets A and B. • A ∩ B = {x | x ∈ A and x ∈ B} • The Intersection can be generalized between the sets in a family of sets. • where {Ai | i ∈ I} is a family of sets The intersection between set A and universal set X is A. A ∩ X = A The intersection of A and empty set is empty set. A ∩ ∅ = ∅ The intersection of A and its complement is all the time empty set. When two sets A and B have nothing in common, the relation is called as disjoint. Namely, it is when the intersection of A and B is empty set. A ∩ B = ∅

9. 1.2.4 Partition of Set Definition (Partition) A decomposition of set A into disjoint subsets whose union builds the set A is referred to a partition. Suppose a partition of A is π, π(A)= {Ai | i ∈ I, Ai ⊆ A} then Ai satisfies following three conditions. (1) ∅≠iA(2) , ∅=∩jiAAji, ≠Iji∈, (3) □ AAIii=∈UIf there is no condition of (2), π(A) becomes a cover or covering of the set A.

10. 1.4 Definition of fuzzy set 1.4.1 Expression for fuzzy set Membership function μA in crisp set maps whole members in universal set X to set {0,1}. μA : X →{0, 1} Definition (Membership function of fuzzy set) In fuzzy sets, each elements is mapped to [0,1] by membership function. μA : X →[0, 1] Where [0,1] means real numbers between 0 and 1 (including 0,1). Consequently, fuzzy set is “vague boundary set” comparing with crisp set.

11. Graphical representation of crisp and fuzzy sets

15. Type-n Fuzzy Set Example of type-2 fuzzy set

16. Level-k fuzzy set Example of level-2 fuzzy set