Analyse 1: exercises avec solutions by E. (Edmond) Ramis, C. (Claude) Deschamps, J Odoux

By E. (Edmond) Ramis, C. (Claude) Deschamps, J Odoux

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26. Let X and Y be two universes of discourse. A fuzzy binary relation R in the Cartesian product X x Y is a fuzzy subset in X χ Υ defined by the membership function μκ: X x Y (x, y) > [0, 1 ] , >μ κ (χ, y ) , χ ε χ , y ε Υ, where the grade of membership relationship between x and y. Example similar" (2-106) μ^(χ, y) indicates the degree of 2. 14. Let X = Y = IR. Then R 4 "x and y are very is a fuzzy binary relation which may be defined by- 49 Concepts and Operations of Fuzzy Subsets y ) = e"~ k < x ~ ?

1H aA aA 0 I I I I — x — I I I I x Aqr1 Aq2 Δ<*3 Fig. 6. 0, 0 . 2 , 1 ) . A = L e t X = { χ χ , χ 2 , x 3 , x 4 , x 5 } and A = ( 0 . 7 , Based on ( 2 - 6 0 ) , ( 2 - 6 1 ) , and t h e o r e m 2 . 4, U 0·Α0 34 Basic Concepts of Fuzzy Set Theory = 1 ( 0 , 0, 0, 0, 1 ) U 0 . 7 ( 1 , 0, 0, 0, 1 ) U 0 . 4 ( 1 , 0 . 2 ( 1 , 1, 0, 1, 1) U 0 ( 1 , 1, 1, 1, 1) = [ m a x ( 0 . 7 , 0 . 4 , 0 . 2 ) , m a x ( 0 . 4 , 0 . 2 ) , 0, 0 . 4, 0 . 2 ) ] = ( 0 . 7 , 0 . 4 , 0, 0 . 2 , 1 ) . Example 2. 7. 1, 0, 0, 1) U L e t A ε J*(X) and i s d e f i n e d by μΑ( χ ) = 1 -" For α ε [0, 1 ] , the characteristic is fA3x oo = ET .

12. Let A, Β ε ^ ( Χ ) . The algebraic sum of A 28 Basic Concepts of Fuzzy Set Theory and B, denoted as A + B, is defined by ^A + β ( χ ) = Mx> + M x > " ^Α^ χ ) * M x > > VxeX. 13. Let A, B ε J*(X). The algebraic A and B, denoted as A · B, is defined by μΑ # B (x) (2-48) product of = μΑ(χ) · μ Β (χ), V x e X . (2-49) Remark. + and · reflect trade-offs between A and B. They can be regarded as interactive operators in defining U and fl. However, + and . satisfy only the law of commutativity, associativity, two-extremes, and De Morgan's Law.

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