By P. F. Hsieh, A. W. J. Stoddart

ISBN-10: 3540053697

ISBN-13: 9783540053699

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**Extra info for Analytic Theory of Differential Equations**

**Sample text**

And we deﬁne the following sets (a) A = {x ∈ Z | for some integery > 0, x = 2y} (b) B = {x ∈ Z | for some integery > 0, x = 2y − 1} (c) A = {x ∈ Z | for some integerx < 10} Describe ∼ A, ∼ (A ∪ B), ∼ C, A− ∼ C, andC − (A ∪ B) 4. Show that for all sets A, B and C (A ∩ B) ∪ C = A ∩ (B ∪ C) iﬀ C ⊆ A 5. What is the cardinalty of {{1, 2}, {3}, 1}. 6. Give the domain and the range of each of the following relations. Draw the graph in each case. com 49 Chapter 4. Sets Mathematics for Computer Scientists (a) {(x, y) ∈ R × R} | x2 + 4y2 = 1} (b) {(x, y) ∈ R × R} | x2 = y2} (c) {(x, y) ∈ R × R} | 0 ≤ y, y ≤ x and x + 1y ≤ 1} 7.

This is a very wide deﬁnition and one that is very similar to that of a relation , the critical point is that for each a there is a unique value y. A common way of writing functions is f:X→Y which illustrates that we have two sets X and Y together with a rule f giving values in Y for values in X. We can think of the pairs (x, y) or more clearly (x, f(x)). This set of pairs is the graph of the function In what follows we show how functions arise from the idea of relations and come up with some of the main deﬁnitions.

The number of customers who have only have a Visa card is? com 56 Chapter 5. Counting Mathematics for Computer Scientists 5. So for example from the New York Times According to a New York Times report on the 16 top-performing restaurant chains (a) 11 serve breakfast. (b) 11 serve beer. e. they server alcohol and all meals. All 16 oﬀered at least one of these services. A total of 5 were classiﬁed as ”family chains,” meaning that they serve breakfast, but do not serve alcohol. Further a total of ﬁve serve breakfast and have full table service, while none serve breakfast, beer, and also have full table service.

### Analytic Theory of Differential Equations by P. F. Hsieh, A. W. J. Stoddart

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