GATE | GATE-CS-2006 | Question 2

Let X, Y, Z be sets of sizes x, y and z respectively. Let W = X x Y. Let E be the set of all subsets of W. The number of functions from Z to E is:
(A) z2xy
(B) z x 2 xy
(C) z2x + y
(D) 2xyz


Answer: (D)

Explanation: Number of functions from a set A of size m to set B of size n is nmn^m, because each of the m elements of A has
n choices for mapping. Now here m=|Z|=zm = |Z| = z, and n=|E|=2xyn = |E| = 2^{xy} because number of subsets of a set of size n is 2n2^n,
and here set W has size of xyxy.

So number of functions from Z to E = (2xy)z=2xyz(2^{xy})^z = 2^{xyz}. So option (D) is correct.

Source: http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2006.html


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