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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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**Fuzzy Relationsand 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 (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**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**Properties of some binary relations**• Reflexive • Anti-reflexive • Symmetric • Anti-symmetric • Transitive • Equivalence • Partial order • Total order • Assignment: Classify: =,<,>,<=,>=**Fuzzy Relations**• Let U and V be universes and let the function • Continuous relations • Discrete relations**“Approximately Equals”**Example 2.50 Universe of Discourse Tabular**Example 2.51: ”Much taller than”**Express the relation as an “integral”**Example 2.52: IF-Then Rule**• Programming If-Then Convert to integral form using two versions of AND**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**Projection**Simple case 1: Case 2: General case**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**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**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**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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