I know that $f: X \to Y$ means that $f(x) = y \in Y$ where $x \in X$.
But what is the difference between $f: X \to Y^2$ and $f: X \to 2^Y$. I know that one of the notations denotes $f(x) = (y_1,y_2)$, but I'm not sure which one of these two.
And finally, what does $R: A \to 2^{W \times W}$ mean when $A$ is a set $A = \{a_1,a_2\}$ and $W$ is a set $W = \{w_1,w_2,w_3\}$?
In analogy with $\mathbb{R}^2$, the Cartesian plane, $Y^2$ is the set of all pairs $(y_1,y_2)$ with $y_1,y_2\in Y$.
On the other hand, $2^Y$ is the set of all functions from $Y$ into $2=\{0,1\}$. (This is sometimes also identified with the power set of $Y$, with a function $\chi: Y \to \{0,1\}$ being identified with $\chi^{-1}(1)$.)
Thus, $R: A \to 2^{W\times W}$ sends a point $a\in A$ to a function $R(a): W\times W \to \{0,1\}$ (alternatively, a subset of $W\times W$). Since $A=\{a_1,a_2\}$, $R$ picks out two subsets of $W\times W$ (of which there are $2^{3\cdot 3}=2^9$ of them).