Create the inverse of mapping

Question

What is the inverse mapping of $f:{\{0,1\}}^{\{0,1\}}\mapsto{\{0,1,2,3\}}, b \mapsto b_0*2^0+b_1*2^1$ ?

For me it is not clear how I can do that with a polynom consisting more than one variable (here are two: $b_0, b_1$).

Answer 1

The elements of $\{0,1\}^{\{0,1\}}$ are functions from $\{0,1\}$ to $\{0,1\}$, which is the same as a two-emelent sequence where the elements are either $0$ or $1$. In other words, there are the four elements $(0,0), (0,1),(1,0),(1,1)$. We have, for instance, $f(0,1) = 0\cdot 2^0 + 1\cdot 2^1 = 2$. Can you do the rest? The inverse can be described by simply listing all the elements of $\{0,1,2,3\}$ and what they are mapped to, like $2\mapsto (0,1)$.