Let $X,Y,Z$ be sets of sizes $x, y$ and $z$ respectively. Let $W = X \times Y$ and $E$ be the set of all subsets of $W$. The number of functions from $Z$ to $E$ is

…………………..

D is Correct.

$|E| = 2^{xy}$ which is the number of subsets of $W.$

Now, the mapping for a function from $A$ to $B$ with $N$ and $M$ elements respectively can be done in $M^{N}$^{ }ways.

Here,

$|E|^z = \{2^{xy}\}^z = 2^{xyz}$

W = X ⨉ Y ,∣ W ∣ = xy

E = powerset of ( X ⨉ Y ) , ∣ E ∣ = 2^{xy}

Let f be the function , f : Z ---> E

Total number of functions from Z ---> E = ∣ E ∣^{ ∣ Z ∣ }= 2^{xyz}