# Thread: Constructing a bijection between a unit interval and a unit square.

1. ## Constructing a bijection between a unit interval and a unit square.

Hi thanks for helping in advance,

Here is the question....

Construct a Bijection between the closed unit interval [0,1] and the closed unit square.

[0,1] x [0,1] = {(x,y) : 0 <= x <= 1 and 0 <= y <= 1}.

Any ideas would be appreciated

Thanks again.

2. The definitive answer is to take each (real) number in the interval [0, 1]. Take the even-indexed digits of its decimal expansion to form the x co-ordinate of a point in the square, and the odd-indexed digits to form the y co-ordinate.

That's the overall policy - are you in a position to be able to tighten it up, i.e. prove that this is indeed a bijection?

3. Thanks Matt,

I can certainly see that it is a Bijection. May main problem seems to be how I write that explicitly. Actually that seems to be most of the problem I have in the class. Describing what is in my head.

Probably just takes practice.

An example would be great.

4. Actually it turns out that as stated, it is not a bijection. Since it is not injective.

Consider cases: .71707070707....

and .707979797979797.....

Clearly different numbers but both map to the same place. namely (.7777777...., .1 )

So you have to tweak it to make it work.