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 µ A (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}. µ A : 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 {A i | 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 {A i | 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}. μ A : X →{0, 1} Definition (Membership function of fuzzy set) In fuzzy sets, each elements is mapped to [0,1] by membership function. μ A : 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