36 votes 36 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 $z^{2^{xy}}$ $z \times 2^{xy}$ $z^{2^{x+y}}$ $2^{xyz}$ Set Theory & Algebra gatecse-2006 set-theory&algebra normal functions + – Rucha Shelke asked Sep 16, 2014 • edited Jun 21, 2017 by Silpa Rucha Shelke 7.1k views answer comment Share Follow See all 2 Comments See all 2 2 Comments reply Mohitdas commented Dec 18, 2021 reply Follow Share ………………….. 3 votes 3 votes ananya_23 commented Sep 15, 2023 reply Follow Share this clears it up. thanks! 1 votes 1 votes Please log in or register to add a comment.
Best answer 47 votes 47 votes 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, $|E|^z = \{2^{xy}\}^z = 2^{xyz}$ Snehil Joshi answered Jan 31, 2015 • edited Nov 23, 2017 by pavan singh Snehil Joshi comment Share Follow See all 0 reply Please log in or register to add a comment.
13 votes 13 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) Mayank answered Jan 31, 2015 Mayank comment Share Follow See all 0 reply Please log in or register to add a comment.
6 votes 6 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 Warrior answered Aug 14, 2017 Warrior comment Share Follow See all 0 reply Please log in or register to add a comment.
0 votes 0 votes Given set ==> X Y Z sizes ==> x y z W = X x Y ∣ W ∣ = xy E = powerset of ( X ⨉ Y ) , ∣ E ∣ = 2xy E be the set of all subsets of W Therefore E = 2^(x*y) so number of functions from Z to E will be Z ---> E = ∣ E ∣ ^∣ Z ∣ = 2^xyz akshay_123 answered Sep 19, 2023 • edited Sep 26, 2023 by akshay_123 akshay_123 comment Share Follow See all 0 reply Please log in or register to add a comment.