Fuzzy Logic Lecture Notes

Introduction to Fuzzy Logic

• Fuzzy Logic Definition: Introduced by Lotfi Zadeh in 1965.
  • Deals with imprecise, vague situations where true/false states aren't clear.
  • Offers flexibility in reasoning, similar to human reasoning, accommodating uncertainties.

Crisp Values

• Definition: Precise values with strict true/false boundaries.
  • Boolean System: 1 (true), 0 (false).
  • Fuzzy Logic allows intermediate values, partially true/false.

Example: Car Speed

• Boolean System: Only yes/no answers (e.g., "Is the car moving fast?").
• Fuzzy Logic System: Allows for partially true answers (e.g., "extremely fast," "fast," "slow").

Membership Functions

• Introduced by: Lotfi Zadeh, 1965.
• Purpose: Characterize fuzziness, represent truth degree.
  • Boolean Example: Membership value of "no" is 0, "yes" is 1.
  • Fuzzy Example: Values like 0.9 (close to true), 0.1 (close to false).

Introduction to Classical Sets

• Definition: Collection of distinct objects with crisp values (e.g., marks above 75).
• Membership: Only members (elements above threshold) and non-members.
  • Membership Function: Chi (χ), 1 if element in set, 0 otherwise.
• Cardinality: Number of elements in a set.

Fuzzy Sets and Extension of Classical Sets

• Fuzzy Sets: Allow partial membership in a set, varying degrees of membership.
• Comparison:
  • Classical Sets: Precise membership properties.
  • Fuzzy Sets: Imprecise membership properties.

Graphical Representation

• Classical Set Graph: Only values 0 and 1.
• Fuzzy Set Graph: Partial membership values.
• Mathematical Definition: Fuzzy set A in universe U as a set of ordered pairs.
  • Membership function μ, range [0, 1].

Example: Speed

• Speeds mapped on graph, membership values indicate closeness to true/false.
• Different Curves: Trapezoid, triangular, sigmoid, Gaussian functions.

Representation of Fuzzy Sets

• Discrete and Finite Universe: Summation notation for elements.
• Continuous and Infinite Universe: Integration notation for elements.

Example: Differences in Representation

• Fuzzy Set: Allows for varying membership values.
• Crisp Set: Only full members between defined boundaries.

Fuzzy Set Example

• Two ways of representation: summation and ordered pair notation.

Fuzzy Membership Functions

• Use data or expert opinion to define curves.

Conclusion

• Fuzzy Logic: Provides flexibility in reasoning, accommodating real-world vagueness.
• Crisp Logic: Strict boundaries, fails to represent real-world complexities.

Note: Understand the difference between fuzzifying descriptions and the precise mathematical functions used in fuzzy sets.