Total number of Possible Functions

Let A and B be two sets with m and n elements, respectively. A function from A to B assigns each element of A to exactly one element of B. Since each element of A has n possible choices in B, the total number of functions from A to B is nᵐ.

For example: X = {a, b, c} and Y = {4, 5}. A function from X to Y can be represented in the figure below.

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

Number of onto functions from one set to another

Let X and Y be two finite sets with ∣X∣ = m and ∣Y∣ = n. A function f : X → Y assigns each element of X to exactly one element of Y.

• For each element of X, there are n choices in Y.
• Since there are m elements in X, the total number of functions is:

Note:

• 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.

Solved Question

Question 1: 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: 

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) 

Question 2: 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 Log₂(Log₂N) is: 

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.

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

Solution:

Using m = 4 and n = 3, the number of onto functions is: 
3⁴ - ³C₁(2)⁴ + ³C₂1⁴ = 36. 

Practices Problems

Question 1: How many functions can be defined from a set with 3 elements to a set with 2 elements?

Question 2: If set A has 4 elements and set B has 3 elements, how many functions can be defined from A to B?

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

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

Question 5: 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?

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