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)z2xyb)z x 2xyc)z2x + yd)2xyzCorrect answer is option 'D'. Can you explain this answer?

Question

Answer

Explanation:

To find the number of functions from set Z to set E, we need to determine the number of possible mappings from each element in Z to a subset of W.

Step 1: Finding the size of set W
The set W is defined as the Cartesian product of sets X and Y. The size of W is given by the product of the sizes of X and Y, i.e., |W| = x * y.

Step 2: Finding the number of subsets of W
The set E is defined as the set of all subsets of W. The number of subsets of a set is given by the formula 2^n, where n is the size of the set. Therefore, the number of subsets of W is 2^|W| = 2^(x * y).

Step 3: Finding the number of functions from Z to E
For each element in Z, we can choose any subset of W as its mapping. Since there are 2^(x * y) subsets of W, there are 2^(x * y) choices for each element in Z.

As there are z elements in set Z, the total number of functions from Z to E is given by the product of the choices for each element, i.e., 2^(x * y)^z.

Simplifying the expression, we get (2^(x * y))^z = 2^(x * y * z).

Therefore, the correct answer is option 'D', which is 2^(x * y * z).

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