By Keith B. Oldham
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Extra resources for The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order
Example text
Ln=1,2, ... }u{0}; (c) X is a nonempty set, E is a nonempty subset of X, C({ = {F lEe F eX}; (d) X is a nonempty set, E is a nonempty subset of X, E ¢ X, C({ = {FIF c E}; (e) X is a nonempty set, E is a nonempty subset of X, C({ = {E}. 15. 14? 16. }, where A = [1 - lin, n], n = 1,2, ... }, n = 1,2, ... 17. 16, which classes are closed under the formation of arbitrary intersections? Verify that d[C€] c C€ for these classes C€. 18. 16, which classes are AU-classes? 21 is applicable in some of these cases.
6 (4), (5), and (6). 10. Let X = (-00,00) x (-00,00) = {(x, y) 1-00 < x < 00, -00 < y < oo}. Prove that the class of all sets that have the form is a semiring. 11. 11. 12. Is a monotone class closed under the formation of limit operations of set sequence? Why or why not? 13. Prove that fFp(C({) = {n U tET Es IEs E C({, S, and T are arbitrary index sets}. 14. }, C({={{n,n+1, ... }ln=1,2, ... }u{0}; (c) X is a nonempty set, E is a nonempty subset of X, C({ = {F lEe F eX}; (d) X is a nonempty set, E is a nonempty subset of X, E ¢ X, C({ = {FIF c E}; (e) X is a nonempty set, E is a nonempty subset of X, C({ = {E}.
I E, E, 'ET E C€, t E T, T is an arbitrary index set}, then d[ C€'] = d[ C€]. Can you find a class larger than C€' for which this result still holds? 16. Prove that any set in C€ may be expressed by an intersection of the holes of C€, moreover, prove that any union of sets in C€ may be expressed by an intersection of the holes of C€. 21. 'e[ C€], T is an arbitrary index set}. 25. 'e[C€] c C€. 26. 16, which classes are closed under the formation of unions? 'e[C€] c C€ for these classes. 27. Prove that if C€ is an AI-class, then X e C€.