Question

MTH 182 Work sheet 0S: Bijections and Cardinality of Setav tTa pti: The triangle of positive rational numbers below is called the Stern Broor tree. The e tree consists of all reduced positive rational numbers and is generated by evaluating all possible finite equences of com )(1/1) such that each of the functions fk in the compositions is positions (f 。左. 。1。 one of the two the functions α and β defined by where alb e O* is reduced (ged(a, b) 1 and a,b E N). Thus, for example, (α。β)(1/1)=2/3, and (β°a)( 1 / 1 ) = 312. (or ° α)( 1 / 1)=1/3, (a) Find the unique sequences of compositions of α and β whose values at l / l are 4/7 and 7 /4 (b) Find the unique sequence of compositions of a and ß whose value at 1/1 is 55/38. c) Show that a and B are bijections. Hint: Express a and ß in terms of q alb and then find their inverses!

Answer 1

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