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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

1. $z^{2^{xy}}$
2. $z \times 2^{xy}$
3. $z^{2^{x+y}}$
4. $2^{xyz}$

edited | 1.8k views

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,

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

by (279 points)
edited
çardinality of set W = x*y

and E is set of all subsets of W which means E is power set of W so cardinality of set E = 2^(x*y)

now we have cardinalities of both E and Z so number of functions from Z to E will be (cardinality of E)^(cardinality of Z) so ans is 2^(x*y*z)
by (123 points)

W = X  ⨉ Y   ,∣ W ∣ = xy

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

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

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

## The correct answer is ,(D) 2xyz

by Loyal (8.1k points)
ans (D)
by Active (5k points)
reshown