# Differential Equations with Mathematica by M. Abell, J. Braselton By M. Abell, J. Braselton

Differential Equations with Mathematica thoroughly covers developing, numerical computing, and approximating ideas to boring and partial differential equations. The ebook serves as a hands-on creation to the subject material via a variety of examples that specify find out how to resolve vital purposes utilizing Mathematica.

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Axiom C3) C is a continuous function. (Axiom C4) C is involutive. 3). C (a) ­1 for a d t ® ¯0 for a ! t C(a) 1 C(a) = 1 - a 1 a Fig. 1. Standard complement set function 30 2. The Operation of Fuzzy Set PA(x) PA (x) A A 1 1 x 1 1 x Fig. 2. Illustration of standard complement set function C(a) ­1 for a d t C ( a) ® ¯0 for a ! t 1 t a 1 Fig. 3. Example of complement set function Note that it does not hold C3 and C4. 4). 75 in this function. 33, C4 does not hold now. 5). When w = 1, the Yager’s function becomes the standard complement function C(a) = 1 - a.

Axiom I2) I(a, b) = I(b, a), Commutativity holds. (Axiom I3) If a d a’ and b d b’, I(a, b) d I(a’, b’), Function I is a monotonic function. (Axiom I4) I(I(a, b), c) = I(a, I(b, c)), Associativity holds. Just like in the union function, these four axioms are the axiomatic skeleton, and the following two axioms can be added. 36 2. The Operation of Fuzzy Set (Axiom I5) I is a continuous function (Axiom I6) I(a, a) = a, I is idempotency. 2 Examples of Intersection Standard fuzzy intersection completes the above 6 axioms.

With this idea, we can define an operatorଠfor the exclusive OR disjoint sum as follows. P A'B ( x) P A ( x)  P B ( x) Ƒ 40 2. 1 0 x1 x2 x3 x4 Fig. 11. 1 0 x1 x2 x3 x4 Fig. 12. 4 By using standard complement and intersection operations, the difference operation would be simple. 7), (x3, 0), (x4, 0)} Ƒ B A Fig. 13. 2 x1 x2 x3 x4 Fig. 14. Simple difference A – B 42 2. 1 x1 x2 x3 x4 Fig. 15. 15). 3 Distance in Fuzzy Set The concept ‘distance’ is designated to describe the difference. 16). Measures for distance are defined in the following.