This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MATH 290 – HOMEWORK 5 – SOLUTIONS Definition. An ordered triple of natural numbers ( a,b,c ) is a Pythagorean triple if a < b < c and a 2 + b 2 = c 2 . The set of all Pythagorean triples is denoted by P . The numbers a and b are called the legs of the triple and c is called the hypotenuse . Definition. We say that two triples ( a,b,c ) and ( a ,b ,c ) in P are similar provided that there exist natural numbers l and k such that la = ka , lb = kb and lc = kc . Definition. A point ( x,y ) on the unit circle (that is, such that x 2 + y 2 = 1) is said to be a rational point if x and y are rational numbers with 0 < x < y . The set of all rational points is denoted by R . 1. Prove that similarity as defined above is an equivalence relation on P . Solution: Note that ( a,b,c ) is similar to ( a,b,c ) by taking l = k = 1 in the definition. Hence the relation is reflexive. If ( a,b,c ) is similar to ( a ,b ,c ) with la = ka , lb = kb and lc = kc , then ( a ,b ,c ) is similar to ( a,b,c ) with ka = la , kb = lb and kc = lc . Hence the relation is symmetric. Suppose that ( a,b,c ) is similar to ( a ,b ,c ) with la = ka , lb = kb and lc = kc , and that ( a ,b ,c ) is similar to ( a 00 ,b 00 ,c 00 ) with ma = na 00 , mb = nb 00 and mc = nc 00 . Then mla = mka = nka 00 , and similarly mlb = nkb 00 and mlc = nkc 00 . Hence ( a,b,c ) is similar to ( a 00 ,b 00 ,c 00 )....
View
Full
Document
This note was uploaded on 03/05/2011 for the course MATH 290 taught by Professor Alligood,k during the Fall '08 term at George Mason.
 Fall '08
 Alligood,K
 Advanced Math, Natural Numbers

Click to edit the document details