Surjective function


In the area of ​​mathematics, a function is a relationship that exists between a given set X, also known as the domain name, and another set of elements Y, known as the codomain , in such a way that each element x of the domain corresponds to a single element f (x) of the codomain. The function, defined in simpler and simpler language is a common logical process that is expressed as “depends on” . The functions can be classified depending on the relationship between the elements that the domain has , the codomain and image . Mathematical functions in real life can refer to everyday situations, such as the cost of a phone call that depends on its duration, or the cost of sending an order that depends on its weight. A function is a relationship between two sets in which each element of the first set corresponds to a single element of the second set. A surjective function , also known as surjective , epjective, or surjective function is a function in which each result value has at least one source value .


What is a surjective function?

We say that a function is surjective or surjective if all the elements that are in the image, AND have anti-image . In other words, if for any y of the image Y there exists at least one element x of the image such that f (x) = y.

  • Definition of surjective function
  • Properties of the surjective function
  • Applications
  • How to know if it is a surjective function
  • Examples

Definition of surjective function

A function is surjective, surjective or exhaustive, it occurs when the codomain and the path coincide. Formally, it can be defined as follows:


∀y  ∈ Cod f  ∃ x  ∈ Dom f | f (x) = y

This means, for any element y for the codomain there is another element x of the domain such that y is the image of x times f. In mathematics, a function is surjective if it is applied over the entire codomain , that is, when each element of “Y” is the image of at least one element of “X”. To make it a little easier to understand, we can say that each set or starting element has to cover the elements of the codomain or arrival set . Each value in a surjective function has to have its respective partner in the target set to be a surjective function.

Properties of the surjective function

Among the properties of the surjective function we mention the following:
It has domain of the function which is the set of values ​​that the independent variable can take, that is, those values ​​for which the function is defined.

Regarding the image of the function, it has growth , which includes the increasing that occurs if when x increases, y is also increased; and decreasing , what happens if when increasing the value of x, the value of y. It also has a constant, if by varying x, y remains the same.


In our daily life, the surjective function can and is applied in different daily situations . For example, it is used in optimization problems in the statements of multiplicative structure problems or of the traveling agent, where all the nodes that go from side 1 to side 2 have to be occupied. In the area of finance when to each investment portfolio corresponds to one or more investors.

Imagine, for example, that we are in a warehouse or a supermarket where there are many different products , each product found in the supermarket has a unique barcode and that only belongs to that product. The starting point here are the products and the arrival sets will be the barcodes. This means that for a product that is in the supermarket, there is only one barcode.

How to know if it is a surjective function

We know that a function is surjective or surjective when each of the elements that exist within the range is the image of at least one element of the domain of said function. Another way to determine when a function is surjective is if every element of the final set Y has at least one element of the initial set X to which it corresponds. This means that a function is surjective if the path of the function is the final set Y.

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