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# assn1 - Math 361 Problem Set 1 1(1.2.9 If C 1 C 2 C 3 are...

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Unformatted text preview: Math 361, Problem Set 1 August 27, 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. 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) 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 )....
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assn1 - Math 361 Problem Set 1 1(1.2.9 If C 1 C 2 C 3 are...

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