Mathematics | Total number of Possible Functions

Last Updated : 12 Aug, 2024

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

Number of functions from one set to another: Let X and Y be 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, the 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) 16

Solution: 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 2¹⁶. Therefore, S has 2¹⁶ elements. Also, given, N denotes the number of function from S(2¹⁶ elements) to {0, 1}(2 elements). Therefore, N has 2²¹⁶ elements. Calculating required value,

Log2(Log2 (2²¹⁶)) =Log2¹⁶ = 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, the total number of functions is 2^m. Out of these functions, 2 functions are not onto (If all elements are mapped to 1^st element of Y or all elements are mapped to 2^nd 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,
[Tex]n^{m}-{n \choose 1}(n-1)^{m} +{ n \choose 2}(n-2)^{m}……..(-1)^{n-1}{n \choose n-1}1^{m} [/Tex]

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:
3⁴ – ³C₁(2)⁴ + ³C₂1⁴ = 36.

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Practices Problems – Total number of Possible Functions

1. Basic Function Counting

– How many functions can be defined from a set with 3 elements to a set with 2 elements?

2. Function from Set to Set

– If set A has 4 elements and set B has 3 elements, how many functions can be defined from A to B?

3. Injective Functions

– How many injective (one-to-one) functions can be defined from a set with 3 elements to a set with 5 elements?

4. Surjective Functions

– How many surjective (onto) functions can be defined from a set with 4 elements to a set with 3 elements?

5. Functions with Constraints

– If there are 4 elements in set A and 3 elements in set B, and each element in B must be the image of exactly 2 elements in A, how many such functions are possible?

6. Functions with Identical Sets

– How many functions can be defined from a set with 3 elements to itself?

7. Function Counting with Restrictions

– Consider a function from a set with 4 elements to a set with 2 elements. How many functions exist where each element of the codomain is mapped to by at least one element from the domain?

8. Bijective Functions

– How many bijective (one-to-one and onto) functions are there from a set with 3 elements to another set with 3 elements?

9. Function Counting with Specific Range

– How many functions can be defined from a set with 2 elements to a set with 4 elements if the range of the function must include exactly 2 elements?

10. Combinations and Functions

– If set A has 5 elements and set B has 4 elements, how many functions from A to B are there if exactly 3 elements of B must be used as the image of at least one element of A?

Discrete Maths | Generating Functions-Introduction and Prerequisites

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Mathematics | Total number of Possible Functions

Practices Problems – Total number of Possible Functions

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