In this article, we are discussing how to find number of functions from one set to another. For understanding the basics of functions, you can refer this: Classes (Injective, surjective, Bijective) of Functions.

**Number of functions from one set to another –** Let X and Y are two sets having m and n elements respectively. In a function from X to Y, every element of X must be mapped to an element of Y. Therefore, each element of X has ‘n’ elements to be chosen from. Therefore, total number of functions will be n×n×n.. m times = n^m.

For example, X = {a, b, c} and Y = {4, 5}. A function from X to Y can be represented in Figure 1.

Considering all possibilities of mapping elements of X to elements of Y, the set of functions can be represented in Table 1.

**Examples –** Let us discuss gate questions based on this:

- Q1. 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) z^{2xy}

(B) z x 2^{xy}

(C) z^{2x + y}

(D) 2^{xyz}

Solution: As W= X x Y is given, number of elements in W is xy. As E is the set of all subsets of W, number of elements in E is 2^{xy}. The number of functions from Z (set of z elements) to E (set of 2^{xy} elements) is 2^{xyz}. So the correct option is (D)

- Q2. Let S denote the set of all functions f: {0,1}^4 -> {0,1}. Denote by N the number of functions from S to the set {0,1}. The value of Log2Log2N is ______.

(A) 12

(B) 13

(C) 15

(D) 16Solution: As given in the question, S denotes the set of all functions f: {0,1}^4 -> {0,1}. The number of functions from {0,1}^4 (16 elements) to {0,1} (2 elements) are 2^16. Therefore, S has 2^16 elements. Also, given, N denotes the number of function from S(2^16 elements) to {0,1}(2 elements). Therefore, N has 2^2^16 elements. Calculating required value,

Log2(Log2 (2^2^16)) =Log2^16 = 16

Therefore, correct option is (D).

**Number of onto functions from one set to another –** In onto function from X to Y, all the elements of Y must be used. In the example of functions from X = {a, b, c} to Y = {4, 5}, F1 and F2 given in Table 1 are not onto. In F1, element 5 of set Y is unused and element 4 is unused in function F2. So, total numbers of onto functions from X to Y are 6 (F3 to F8).

- If X has m elements and Y has 2 elements, the number of onto functions will be 2^{m}-2.
**Explanation:**From a set of m elements to a set of 2 elements, total number of functions is 2^m. Out of these functions, 2 functions are not onto (If all elements are mapped to 1st element of Y or all elements are mapped to 2nd element of Y). So, number of onto functions is 2^{m}-2. - If X has m elements and Y has n elements, the number if onto functions are,

n^m – nC1(n-1)^m + nC2(n-2)^m….. +(-1)^(n-1) nCn-1(1)^m

**Important notes –**

- The formula works only if m>=n.
- If m < n, the number of onto functions is 0 as it is not possible to use all elements of Y.

Q3. The number of onto functions (surjective functions) from set X = {1, 2, 3, 4} to set Y = {a, b, c} is:

(A) 36

(B) 64

(C) 81

(D) 72

Solution: Using m = 4 and n = 3, the number of onto functions is:

3^4 – 3C1(2)^4 + 3C2(1)^4 = 36.

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