Membership Functions in Fuzzy Logic

Introduction

• Discussion on membership functions in fuzzy logic.
• Explanation of fuzzy sets and their differences from crisp sets.

Crisp Sets vs. Fuzzy Sets

• Crisp Set:
  • Elements are either present or not present.
  • Example: Set A = {1, 2, 3, 4}
    • Element 5 is not in set A.
• Fuzzy Set:
  • Introduction of partial membership.
  • Elements can have a degree of membership.
  • Example of Partial Membership:
    • Element 1: Membership = 0.6 (60% present, 40% not present)
    • Element 2: Membership = 0.3 (30% present, 70% not present)
  • Membership values range from 0 to 1.

Membership Function

• Definition: A curve that defines how each point in the input space is mapped to membership values between 0 and 1.
• Representation:
  • X-axis: Elements of the fuzzy set.
  • Y-axis: Membership values (0 to 1).
• Formal Definition:
  • A fuzzy set A in a universe of discourse X represented as a pair (element, membership value).
  [ ext{Fuzzy Set } A = ext{(Element , } \mu x ext{ of } A) ]
  • Where ( \mu x ) is the membership function.

Features of Membership Functions

1. Core:

   • Region characterized by complete membership (membership value = 1).
   • Example: Elements with membership value of 1.

2. Support:

   • Region characterized by non-zero membership values.
   • Elements whose membership values are greater than 0 but less than 1.
   • A fuzzy singleton: A set where one element has a membership value of 1.

3. Boundary:

   • Region containing elements with non-zero but not complete membership (0 < membership < 1).
   • Elements in the boundary have membership values greater than 0 and less than 1.
   • Mathematical Calculation:
     • Boundary = Support - Core.

Conclusion

• Summary of fuzzy sets, membership functions, and their features.
• Encouragement to like, share, and subscribe for more content.