For any $n \in \mathbb{N}$, let {0, 1, 2}$^n$ = { $(a_1, a_2, ..., a_n)$ | for all $i \in \mathbb{N}_n, a_i \in$ {0, 1, 2} }. Construct a bijection from {0, 1, 2}$^n$ to {$(A,B) | A,B \subset \mathbb{N}_n$ and $A,B$ are disjoint}.
Firstly, I'm a little confused with the first sentence's definition of {0, 1, 2}$^n$. If I take $n=1$, for example, then I would have { $(a_1)$ | for all $i \in \mathbb{N}_1, a_i \in$ {0, 1, 2} }. In other words, the set could be {0} OR {1} OR {2}, as opposed to {0, 1, 2}, right?
Secondly, what does $(A,B)$ mean? Is it referring to the Cartesian product $A$ x $B$?
According to the definition you've given, $\{0,1,2\}^1 = \{(0), (1), (2)\}$. For example, $(0)$ is an ordered $1$-tuple. Similarly $$\{0,1,2\}^2 = \{(0,0), (0,1), (0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)\}$$ Note that this is the same definition as $\mathbb{R}^n$, for example, but where $\mathbb{R}$ has been replaced by a three-element set.
$(A, B)$ is the ordered pair whose first component is $A$ and whose second component is $B$. For example, $(3, 5)$ is a member of $\{(n, m) : n, m \in \mathbb{N}\}$. You can implement $(a, b)$ in set theory as $\{a, \{a, b\}\}$ if you like, but that's usually an irrelevant detail of how to coerce a set theory into talking about objects like tuples that are actually interesting.