Chapter14.pdf - Chapter 14 Solutions to Review Questions 14.1 1 0.75 0.5 0.25 0.05 0-1 0 1 2 3 4 5 6 7 8 9 Support Set of A Supp A ={x \u2212 1 < x < 9

Chapter14.pdf - Chapter 14 Solutions to Review Questions...

• 9

This preview shows page 1 - 4 out of 9 pages.

Chapter 14 Solutions to Review Questions 14.1 1 0.75 0.5 0.25 0.05 0 -1 0 1 2 3 4 5 6 7 8 9 Support Set of A { } 9 x 1 x A Supp < < = Note: The above membership function is one of the possibilities for the fuzzy set “Numbers close to 4”. 14.2 Graphical Illustration of extension principle for continuous functions. A B y x 1 1 B i (x) A i (x) f(x)
14.3 A Subnormal Convex Fuzzy Set cuts of A is convex for all E (0, 1) (adapted from Klir and Yuan pp. 23 figure 1.9) A normal non-fuzzy convex set (pp. 24 figure 1.10) Note: Convexity of a fuzzy set does not mean that the membership function of a convex fuzzy set is a convex function. 1 x 0 A(x) A 0 x A(x) 1
( -Cuts should be convex for all E(0, 1]) 14.4 An -cut of a fuzzy set A is a crisp set A that contains all the elements of the Universal set X whose memberships grades in A are greater than or equal to the specified value of , unlike strong -cut which contains elements whose membership grades in A are only greater than . 14.5 ( ) ( ) 2 2 4 x 1 1 B 2 x 1 1 A + = + = Union ( ) ( ) R x 4 x 1 1 , 2 x 1 1 max 2 2 + + Intersection ( ) ( ) R x 4 x 1 1 , 2 x 1 1 min 2 2 + + -Cuts = 0.4 For A: ( ) ( ) ( ) ( ) ( ) { } 2247 . 3 x x A 2247 . 3 x 2247 . 1 2 x 5 . 1 2 x 1 4 . 0 1 2 x 4 . 0 1 2 x 1 4 . 0 2 x 1 1 2 2 2 2 2 = + + α For B: (Using Similar Calculations) (x - 4) 1.2247 x 5.2247 { } 2247 . 5 x x B x = 14.6 function

You've reached the end of your free preview.

Want to read all 9 pages?

• Summer '18
• Shiv
• Convex function, Fuzzy set

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern