 # Surjective Function

Between sets A and B, a surjective function is constructed such that every element of set B is related with at least one element of set A. A surjective function’s domain and range are both the same.

Let’s take a closer look at the surjective function, its features, and some instances.

## What Is a Surjective Function?

Every element of the range set is a co-domain, hence the subjective function is defined with reference to the elements of the range set. A surjective function has an image that is the same as its co-domain. A surjective function’s range, co-domain, and image are all the same. We may also argue that a subjective function is an onto function if every y co-domain has at least one pre-image x domain and f(x) = y. Let’s have a look at surjective function in more detail.

If there is at least one an A such that f(a) = b, a function ‘f’ from set A to set B is termed a surjective function. Because they are all mapped from some member of set A, none of the items are left out in the onto function. Consider the following scenario:

Let A = {a1, a2, a3 } and B = {b1, b2 } then f : A →B.:{(a1, b1), (a2, b2), (a3, b2)}

Here in the above example, every element of set B has been utilized, and every element of set B is an image of one or more than one element of set A.

Surjective functions are those in the previous instances of functions that have no residual elements in set B. Every element of set B has been mapped from at least one member of set A in a surjective function. Also, non-surjective functions have items in set B that haven’t been mapped to any element in set A.

## Properties of Surjective Function

A function is considered to be a surjective function only if the range is equal to the co-domain. Here are some of the important properties of surjective function:

• Only if the range of a function equals the co-domain is it regarded a surjective function. Some of the most essential features of surjective function are as follows:Every element in the co-domain will be assigned to at least one element in the domain in a surjective function.
• In a subjective function, a co-domain element can be an image of more than one domain element.
• In a subjective function, the co-domain is equal to the range.A function f: A →B is an onto, or surjective, function if the range of f equals the co-domain of the function f.
• Every element in the co-domain will be assigned to at least one element in the domain in a surjective function.

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