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Fuzzy Relations and Functions

Fuzzy Relations and Functions. By P. D. Olivier, Ph.D., P.E. From Driankov, Hellendoorn, Reinfrank. Classical to Fuzzy Relations. A classical relation is a set of tuples Binary relation (x,y) Ternary relation (x,y,z) N-ary relation (x 1 ,…x n ) Connection with Cross product

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Fuzzy Relations and Functions

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  1. Fuzzy Relationsand Functions By P. D. Olivier, Ph.D., P.E. From Driankov, Hellendoorn, Reinfrank

  2. Classical to Fuzzy Relations • A classical relation is a set of tuples • Binary relation (x,y) • Ternary relation (x,y,z) • N-ary relation (x1,…xn) • Connection with Cross product • Married couples • Nuclear family • Points on the circumference of a circle • Sides of a right triangle that are all integers

  3. Characteristic Function • Any set has a characteristic function. • A relation is a set of points • Review definition of characteristic function • Apply this definition to a set defined by a relation

  4. Properties of some binary relations • Reflexive • Anti-reflexive • Symmetric • Anti-symmetric • Transitive • Equivalence • Partial order • Total order • Assignment: Classify: =,<,>,<=,>=

  5. Fuzzy Relations • Let U and V be universes and let the function • Continuous relations • Discrete relations

  6. “Approximately Equals” Example 2.50 Universe of Discourse Tabular

  7. Example 2.51: ”Much taller than” Express the relation as an “integral”

  8. Example 2.52: IF-Then Rule • Programming If-Then Convert to integral form using two versions of AND

  9. Operations on Fuzzy Relations • R = “x considerably larger than y” • S = “y very close to x” • Intersection of R and S (T-norms) • Union of R and S (S-norms) • Projection • Cylindrical extension

  10. Projection Simple case 1: Case 2: General case

  11. Example 2.60

  12. Example 2.58 • Each x is assigned the highest membership degree from all tuples with that x • Projections reduce the number of variables • Extensions increase the number of variables

  13. Cylindrical Extension • Extension from 1 D to 2 D Extension form 2D to 3 D proj ce(S) on V = S ce(proj R on V) <>R

  14. Composition • Combines fuzzy sets and fuzzy relations with the aid of cylindrical extension and projection. Denoted with a small circle. • Draw picture of composition of functions • Intersection can be accomplished with any T norm • Projection can be accomplished with any S norm

  15. Extension Principle • Allows for the combination of fuzzy and non-fuzzy concepts • Very important • Allows mathematical operations on fuzzy sets • The extension of function f, operating on A1, …, An results in the following membership function for F When f -1 exists. Otherwise, 0.

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