Discrete Mathematics with Graph Theory (3rd Edition) 73

Discrete Mathematics with Graph Theory (3rd Edition) 73 -...

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Section 3.3 71 (b) This follows immediately from part (a) since (0,1] = (0,1) U {I}. 25. We employ a concept known as stereographic projection. Imagine the sphere sitting on the Cartesian plane with south pole at the origin. Any line from the north pole to the plane punctures the sphere at a unique point and the collection of such lines establishes a one-to-one correspondence between the points of the plane and the sphere except for the north pole. A small modification of this correspon- dence finishes the job. Suppose PO,Pl,P2, ... are the points of the sphere which correspond to the points (0,0), (1, 0), (2,0), ... in the plane; thus, the line from the north pole to (n,O) punctures the sphere at Pn (in particular, Po = (0,0». Map the north pole to (0,0), the origin to (1,0), Pl to (2,0), and so forth and let all other points of the sphere go to the same points "as before. 26. [BB] We are given that A = {al.a2, ... ,an} for some n E N and that B = {bl ,b2,b3, . . . }. Then A U B is countably infinite because it is infinite and its elements can be listed as follows:
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