540hw2ans - HW2 1 1 Prove that any set of triangles in R 2...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 6

540hw2ans - HW2 1 1 Prove that any set of triangles in R 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online