3. Suppose that A, B C R².(a) If A and B are homeomorphic, are A and B homeomorphic?(b) If A and B are connected and homeomorphic, are Aº and Bº homeomorphic?

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Answered: 3. Suppose that A, B C R2. (a) If A and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Exercise 3.3.15. Prove or find a counterexample to the following statement. Let A, B and C be sets. Then (A-B) UC = (AUBUC) - (ANB).
This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS = {(x, y) E AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =
6) Let A = {4,5,6} and B = {3,6,7}. Define relations U, V, and W from A to B as follows, For all (x, y) E A×B, (x, y) eU x>y (x, y) e V > X-y is an integer. W = {(4,7), (6,3), (5,6)} Indicate whether any of the relations U, V, and W are functions from A to B.

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3. Suppose that A, B C R2. (a) If A and B are homeomorphic, are A and B homeomorphic? (b) If A and B are connected and homeomorphic, are Ac and Be homeomorphic?