模糊数学pptChSec---.pptVIP

CH2 Decomposition Theorem, Representation Theory and Extension Principle §2.1 Cut sets of fuzzy sets e.g. 2.1.1 In an exam about “winner”, marks of ten examinees are as follows. Examinee x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Grade (mark)100 94 32 61 85 55 25 72 86 40 Now we pick out winters according to principle “enrolling only those who are outstanding”. Pick out elements which membership degree satisfies Fuzzy set Crisp set Definition 2.1.1 Let A∈F (X) and α∈[0,1]. —— α-cut set of fuzzy set A or α-level set of A; —— strong α-cut set or strong -level set； α —— threshold value or belief level. e.g. 2.1.2 Let us consider example1.2.3 again: A=“round”= e.g. 2.1.3 Let A∈F ( ) and Theorem 2.1.1 Let A,B∈F (X) and α,β∈[0,1], then (1) (2) e.g. Theorem 2.1.1 Let A,B∈F (X) and α,β∈[0,1], then (1) (2) Proof Only prove (2) “?” If A?B, then for all α∈[0,1], Namely (2) Theorem 2.1.4 Let A,B∈F (X), then (1) (2) Proof The other identities can be proved by similar method. Theorem 2.1.6 Let and A∈F (X), then (2) Proof (2) The second identity can be proved by similar method. Generally Example 2.1.5 Let Then And Therefore §2.2 Decomposition Theorem (P56) e.g. 2.1.1 1 0 kerA suppA Definition 2.2.1 Let A∈F (X) and α∈[0,1]. We define a fuzzy set αA with membership as αA is said scalar product (数积) α and A. If A is a crisp set, then Theorem 2.2.2 Let A∈F (X), then Theorem 2.2.2 Let A∈F (X), then Proof Corollary 2.2.1 Let A∈F (X), then for all x∈X Proof x的隶属度=包含x的所有截集中α的最大值。 e.g. 2.2.1 Let and for all α∈[0,1]