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assn1-solns

# assn1-solns - Math 361 Problem Set 1 Solutions 1(1.2.9 If...

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Math 361, Problem Set 1 Solutions September 10, 2010 1. (1.2.9) If C 1 , C 2 , C 3 , . . . are sets such that C k C k +1 , k = 1 , 2 , 3 , . . . , , we define lim k →∞ C k as the intersection k =1 C k = C 1 C 2 . . . . Find lim k →∞ C k for the following, and draw a picture of a typical ’ C k ’ on the line or plane, as appropriate: a. C k = { x : 2 - 1 /k < x 2 } , k = 1 , 2 , 3 , . . . . b. C k = { x : 2 < x 2 + 1 k } , k = 1 , 2 , 3 , . . . . c. C k = { ( x, y ) : 0 x 2 + y 2 1 k } , k = 1 , 2 , 3 , . . . . Note: In addition to the book problem, I ask for a picture of the set. Answer: For ( a ), note that 2 C k for every k , but 2 - is not in every C k for any > 0. This is because if k > 1 , 2 - C k . Therefore lim k →∞ C k = k =1 C k = { 2 } . For ( b ) note that 2 C k for any k . As before, 2 + is not in C k for k > 1 / . Therefore lim k →∞ C k = . 2. (1.2.4) Let Ω denote the set of points interior to or on the boundary of a cube with edge of length 1. Moreover, say the cube is in the first octant with one vertex at the point (0 , 0 , 0) and an opposite vertex at the point (1 , 1 , 1). Let Q ( C ) = C dxdydz . (a.) If C Ω is the set { ( x, y, z ) : 0 < x < y < z < 1 } compute Q ( C ). Describe the set C in words (or picture). (b.) If C Ω is the set { ( x, y, z ) : 0 < x = y = z < 1 } compute Q ( C ). Describe the set C in words (or picture).

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