Math 403: Homework 1

Book problems: 1.2, 1.4, 1.6, 1.8, 1.12
and the following two extra problems:
P1. Finish the proof of the Parallel Postulate from class by showing that if C is a point not on the line l_A,B and l is a line containing C but not intersecting l_A,B, then l must be the line constructed in class (that is, points of the form C+t(B-A)). Hint: Let E be any point on l other than C, and show that the vector C-A cannot be written in terms of the vectors B-A and E-C.

P2. Show that if C and D are distinct points on l_A,B, then l_C,D=l_A,B. Conclude that if two lines l₁ and l₂ intersect in at least two points, they must be the same line.

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Department of Mathematics College of Liberal Arts and Sciences University of Illinois at Urbana-Champaign 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Department Main Office Telephone: (217) 333-3350 Fax (217) 333-9576