Question

Worksheet 05: Bijections and Cardinality of SetsV1 MTH 182 3. (12 pt): The triangle of positive rational numbers below is called the Stern-Brocot tree. 4 4 4 4 aluating all possible finite The tree consists of all reduced positive rational numbers and is generated by ev sequences of compositions (iof2 of-1 f)(1/1) such that each of the functions ft in the compositions is one of the two the functions α and β defined by β(a/b)-(a+b)/b 1 and a,b E N). Thus, for example, a(a / b) :: a/(a+b), where alb E Q+ is reduced (gcd(a, b) (a) Find the unique sequences of compositions of α and β whose values at 1/l are 4/7 and 7/4. (b) Find the unique sequence of compositions of α and β whose value at 1/l is 55 /38. (c) Show that α and β are bijections. Hint: Express α and β in terms o q = a lb and then find their inverses!

Answer 1

ん 3 o» an So, aio bi(action