in Set Theory & Algebra edited by
31 votes

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. $2^{xyz}$
in Set Theory & Algebra edited by

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3 Answers

42 votes
Best answer

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.


$|E|^z = \{2^{xy}\}^z = 2^{xyz}$

edited by
12 votes
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)
5 votes

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


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