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Unformatted text preview: HW2 1 1. Prove that any set of triangles in R 2 whose vertices have rational coor- dinates is countable. Solution. Let T be a triangle with vertices ( r 1 ;r 1 ) ; ( r 2 ;r 2 ) and ( r 3 ;r 3 ) , where r j ;r j 2 Q , j = 1 ; 2 ; 3 . We can always assume that to every triangle T there corresponds a unique triple ( r 1 ;r 1 ) ; ( r 2 ;r 2 ) and ( r 3 ;r 3 ) (just consider the left lower vertex as the &rst one and start labeling the vertices in the counterclockwise direction). Hence, there is an injection from the set of triangles with rational coordinates into Q 2 & Q 2 & Q 2 ¡ N . 2. Consider a set T of letters "T" in the plane without common points (letters "T" can be of di/erent sizes, and they can be rotated). Can such set be uncountable? Justify your answer. Solution. The set T must be countable. Indeed, construct an injective mapping f : T ! Q 2 & Q 2 & Q 2 ¡ N ; where Q 2 = Q & Q ¡ N . Once such map is constructed card ( T ) ¢ card ( N ) , and we are done. The mapping f is constructed as follows. Let T 2 T ; see the sketch below for notations. Assign to " T " an ordered triple of rational points ( P;Q;R ) 2 Q 2 & Q 2 & Q 2 as shown in the sketch (and as in Problem 1): 1 Note to the grader. Please grade the following problems: #2, #3, #5, #8, #9 (20 pts for each problem) 1 Observe that another letter " T " inscribed into the same triangle PQR must intersect the &rst letter " T "; consider two case: the upper part of the second " T " is above the upper part of the &rst " T " and the upper part of the second " T " is below the upper part of the &rst " T ". So, f is an injection, as needed. 3. Consider the line y = x and the set U of letters " & " as shown in the sketch. Notice that each "corner point" of " & " is in the one-to-one correspondence with letter from U . Hence, the set U has the power of continuum, in particular, it is uncountable....
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This note was uploaded on 02/16/2012 for the course MATH 540 taught by Professor Ruan during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08