11 Consider the open interval 01 and let S be the set of poi

11) Consider the open interval (0,1), and let S be the set of points in the open unit square; that is, S = {(x, y) : 0 < x, y < 1}.

(a) Find the 1-1 function that maps (0,1) into, but not necessarily onto, S.

(b) Use the fact that every real number has a decimal expansion to produce a 1-1 function that maps S into (0,1). Discuss whether the formulated function is onto. (Keep in mind that any terminating decimal expansion such as .235 represents the same real number as .234999....)

Solution

Ans-Heres my thought for (b), i define a set [tex]A_n , n\\in\\mathbf{N}[/tex] where n represents the number of decimal places. So n=1 would be one decimal place, n=2 would be two decimal places, and so on. With only one decimal place, [tex]A_1[/tex] contains every combination of (x,y) such that 0 Reference https://www.physicsforums.com/threads/function-mapping-to-open-intervals.177777/