1. ## Constructing injection questions

Let $S_2$ = {0,1}^N, $S_2^*$= { $(a_n) \in S_2$; (there exists $k\in N$) s.t ( $for all n>=k) a_n = 1$}. Recall that the function $f: S_2-S_2^*--> [0,1)$, given by $f((a_n)) = \displaystyle \sum_{n = 1}^{\infty}a_n/2^n$ is a bijection. Denote by $\varphi: [0,1)-->S_2-S_2^*$the inverse function $f^{-1}$. Note that for $x \in [0,1)$,
$\varphi (x)$ is the sequence $\varphi(x) = (\varphi_n(x))_{n = 1}^{\infty}$ $\in S_2-S_2^*$ with $f(\varphi(x)) = \displaystyle \sum_{n = 1}^{\infty} \varphi_n(x)/2^n = x$.

a) Construct an injection $g:$ $S_2-S_2^* --> [0,1) x [0,1)$.

b) Construct an injection $h: [0,1) x [0,1) --> S_2-S_2^*$. (Hint: Interlace decimals)

The question is kind of confusing with all these notations, any help will be useful.

2. ## Re: Constructing injection questions

I am truly not sure what you are asking for?
Do you understand that $S_2$ is the set of infinite bit strings?
That $S_2^*$ is the set of infinite bit strings with at a finite collection of zeros?
Every number in $[0,1)$ has binary representation.
The set $S_2\setminus S_2^*$ represent the binary representation that are unique.

Does that help?

3. ## Re: Constructing injection questions

sorry, I do not understand those terms, I think this question has things to do with decimal reprsentation