An Introduction to Differentiable Manifolds and Riemannian - download pdf or read online

By William M. Boothby

ISBN-10: 0121160513

ISBN-13: 9780121160517

Nice introductory differential geometry textual content! I used this booklet to aid me go my qualifying examination. Yay Boothby!

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Read or Download An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics) PDF

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Extra resources for An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised, Volume 120, Second Edition (Pure and Applied Mathematics)

Example text

Since x e U - (A U B), it follows that x 0 A U B. For x not to be in this union means it may not be in either of the sets. So, x 0 A and x 0 B. Hence, since x e U - A = A and x E U - B = B, it follows that x A (2B. (ii) Pick an arbitrary X E A n B. Then, x e A, so x 0 A. Also, x E B, so x § B. Therefore, x g (A U B), and consequently, x e (A U B). N (b) The proof is left for the reader. Theorem 8 resembles the ways that and and or interact with not (which are also called DeMorgan's Laws in logic).

This turns out to be a very natural way to look at computer circuits. * Exercises 1. Let A ={1, 2, 3 ... , 10), B ={2, 3, 6, 81,and C ={3, 5, 4, 8, 2}. Find the following: (a) BUC (b) BnC (c) B - C (d) A - B (e) A - C 2. Let U={0,1,2,3,4,5, 6,7,8,9), A={0,1,2,3}, {0, 3, 6, 9). (a) FindAUB, AnB, A, (A n B), and (B U C) - A. (b) Find P(A), P(B), 7P(A n B), P(A) n P(B). (c) Is P(A U B) = P(A) U P(B)? Prove your answer. (d) Why doesn't P(A) make sense? 3. Let A = {0, 3)and B = {x, y, z}. Find the following: (a) A x B (b) A x A x B (c) B x A (d) B x A x B B={0,2,4}, and C= 32 CHAPTER 1 Sets, Proof Templates, and Induction 4.

U What do we mean when we say that one proof is analogous to another? In this context, it means that the two proofs have essentially the same logic. Here, for example, one can form the proof of part (ii) from the proof of part (i) by interchanging A and B. The second important set operation, intersection, forms a set from the elements common to two sets. Operations on Sets Definition 2. 17 Let A and B be sets. 6. 6 A n B. Example 2. (a) {1,2, 3} n {3,4,51 = 13). (b) {1, 2, 31 n 14, 5, 6) = 0. (c) N•nZ =N.

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