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

$\displaystyle \varphi (x)$ is the sequence $\displaystyle \varphi(x) = (\varphi_n(x))_{n = 1}^{\infty} $$\displaystyle \in S_2-S_2^*$ with $\displaystyle f(\varphi(x)) = \displaystyle \sum_{n = 1}^{\infty} \varphi_n(x)/2^n = x$.

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

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

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