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Understanding the differences between function and application in mathematics
Understanding the differences between function and application in mathematics
One thing that confused me a lot at the beginning of my studies on set theories was the differences between correspondence, function, and application. This article aims to clarify these concepts in a simple but precise way.
By Robin Pourtaud on

Definition:

Let’s start by correctly defining the notions of function and application in mathematics.

Application

Let and be two sets.

An application is an object that associates for each element an element of .

More formally:

A relation is an application if for every element of is contained in a relation .

In other words:

Let’s visualize this concept!

Diagram

Here is an application from to .

If the concept is still unclear for you:

We define before that a relation is an application if:

Here and so the first condition is verified.

The second condition is also verified because and each element of is associated with exactly one element of . You can deduce that in the case of an application .

Function

Let’s start first by saying that all applications are functions.

Let and be two sets. Let be a subset of . A function is a binary relation that associates for each element an element of .

More formally:

With , a relation is a function if for every element of is contained in a relation .

In other words:

If , then this function is an application.

A function can’t link an element to multiple images!

If associates at least one element with multiple images (i.e., points to several distinct elements at the same time), then we will say that is not a function, but a multivalued function or simply a correspondence.

A multi-valued function is not a function. To learn more.

Diagram

Here is a function from to .

The difference:

We just seen the theory and the original definitions of functions and applications. However, this difference can vary a lot depending on the context. Here are 4 answers to this question:

  1. Synonym: it will often be admitted by abuse of language that the two mean the same thing. It is therefore not necessarily rigorous, but depending on your level, the subject you are working on, the difference may not really matter.
  2. According to the field: without thinking, we will often reserve the term “application” in algebra (“linear application” for example) and that of function in analysis.
  3. Historically: some authors reserve the term function in the case where is a set of integers like for example. [2]
  4. The real answer: What we just defined here!

In summary. An application is a function which is itself a correspondence. The reverse is not true!

To complete this article, here is an example of correspondance.

Diagram

Here is a correspondence from to . This is not a function because is associated with two elements of . This is therefore not an application either.