GATE CS 2006 Q2 — Sets, Relations and Functions

To find the number of functions from set $Z$ to set $E$, we first determine the sizes of the sets involved:

1. Size of $W$: Since $W = X \times Y$, the cardinality of $W$ is: $$|W| = |X| \times |Y| = x \cdot y$$
2. Size of $E$: $E$ is the set of all subsets of $W$ (the power set of $W$). Therefore, the cardinality of $E$ is: $$|E| = 2^{|W|} = 2^{xy}$$
3. Number of functions from $Z$ to $E$: The total number of functions from a domain set $A$ to a codomain set $B$ is given by $|B|^{|A|}$. Here, the domain is $Z$ (with size $z$) and the codomain is $E$ (with size $2^{xy}$). $$\text{Number of functions} = |E|^{|Z|} = \left(2^{xy}\right)^z = 2^{xyz}$$

Thus, the correct option is D.