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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. $2^{xyz}$
asked in Set Theory & Algebra by Active (3.7k points)
edited by | 1k views

4 Answers

+28 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.

Here,

$|E|^z = \{2^{xy}\}^z = 2^{xyz}$

answered by (279 points)
edited by
+9 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)
answered by (115 points)
+3 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

answered by Loyal (6.8k points)
0 votes
ans (D)
answered by Active (5k points)
reshown by


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