870 views

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 | 870 views

D is Correct.

E = 2XY  Which is the total number of subsets of W.

Now, the mapping for a function from A to B with N and M elements respectively... we have $M^{N}$ .

Here,

EZ = 2XY(Z) = 2XYZ

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)

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

ans (D)
reshown