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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 )....
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