Using decimal expansions to map S into (0,1)

[kathrynag]

Homework Statement

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.

Homework Equations

S={(0,1):0<x, y<1}

The Attempt at a Solution

I don't even know where to begin. The whole decimal expansion business has me confused.

 

[kathrynag]

Oops, I made a typo. That (0,1) in S should have been a (x,y).

 

[Mark44]

Just to be clear, S is the quarter plane whose left edge is the y-axis and whose upper edge is the line y = 1, right?

Or did you mean that both x and y are between 0 and 1? If that's what you meant, your description for S should have been written as S = {(x, y) | 0 < x < 1, 0 < x < 1}.

 

[kathrynag]

Just to be clear, S is the quarter plane whose left edge is the y-axis and whose upper edge is the line y = 1, right?

Yeah, that's what I meant.

 

[Mark44]

(.5, .25) is in S, right? .5205 is in (0, 1).

 

[kathrynag]

yeah. Ok, so then we have z=x1y1x2y2...