Surjective Function

The surjective function ensures that every element in set B is matched with at least one element in set A. It means that each item in set B gets mapped to at least one element in set A. 
Example:
If A = {1, 2, 3} and B = {a, b}, and the function is f = {(1, a), (2, b), (3, b)}. Here, both a and b are used. No element in the set B is unmapped, so this is a surjective function. 
Look at the given image to understand how a surjective function works visually. 
 

Functions can be classified into different types based on how elements from one set are mapped to elements of another set. The two main types of functions are: injective functions and surjective functions. The difference between injective and surjective functions is given below:
 

A function is said to be a surjective function only when the range is equal to the codomain. Given are some of the properties of a surjective function:

• In a surjective function, every element of the codomain is mapped by at least one element from the domain. 
• A single codomain element can be the image of more than one domain element.
• The range of a surjective function is equal to its codomain. A function f: A → B is a surjective or onto function if its range is the same as the codomain. 
• Every surjective function has a right inverse, and any function that has a right inverse is surjective. 
   

For finding the number of onto functions from a set A with n elements to a set B with m elements: 
Total number of functions from A to B = mn
Number of surjective functions = Total number of functions - Number of functions that are not surjective.
The number of surjective functions can be calculated by using a special formula.
The formula to find the total number of functions that are not surjective:
1mm - 1n +  2mm - 2n - 3mm - 3n + … - m - 1m1n 
The number of surjective functions that can be found using the formula:
mn - 1mm - 1n + 2mm - 2n - 3mm - 3n + … - m - 1m1n
When n < m, the number of surjective functions = 0
n = m, the number of surjective functions = m!

In real life, we sometimes need to match items from one group to another so that nothing in the second group is left out. This idea is similar to a surjective function. Such functions are useful in many areas, like computer science, business, and daily activities. Some examples include: 

• Assigning Students to Classrooms: A surjective function ensures that no classroom is left empty. When each classroom must have at least one student, we can represent students as the domain and classrooms as the codomain. 
• Distributing Jobs to Employees: In companies, each job or task needs at least one employee to handle it. Employees (domain) are assigned to jobs (codomain). A surjective function ensures all jobs are covered and nothing is left undone.
• Ticket Allocation for Events: When tickets are assigned to seats in a hall for an event, each seat must be booked. In this case, tickets are mapped to seats, and a surjective function ensures that no seat is left empty.
• Website User Access: When designing websites, certain roles or permissions must be assigned to all user groups. Surjective mapping ensures every role is assigned to at least one user.
• Course Assignments in Universities: Each course must have at least one student enrolled for it to be active. Students are mapped to courses, ensuring all courses have participants.