Understanding the differences between function and application in mathematics

Definition:

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

Application

Let $A$ and $B$ be two sets.

An application $f$ is an object that associates for each element $a\in A$ an element $f(a)$ of $B$.

More formally:

A relation $R\subseteq A \times B$ is an application if every element $a$ of $A$ is associated with exactly one element $b$ of $B$.

In other words: $\forall a\in A,\ \exists!b\in B\text{ such that }(a,b)\in R$.

Let’s visualize this concept!

Follow the moving current to read the direction of each edge.

Here $R = \{(X,1), (Y,1)\}$ is an application from $A$ to $B$.

If the concept is still unclear for you:

We define before that a relation $R$ is an application if:

1. $R \subseteq A \times B$
2. $\forall a\in A,\ \exists!b\in B\text{ such that }(a,b)\in R$

Here $A \times B = \{(X,1), (X,2), (Y,1), (Y,2)\}$ and $R = \{(X,1), (Y,1)\}$ so the first condition is verified.

The second condition is also verified because $A = \{X,Y\}$ and each element of $A$ is associated with exactly one element of $B$. You can deduce that in the case of an application $|R| = |A|$.

Function

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

Let $A$ and $B$ be two sets. Let $A'$ be a subset of $A$. A function $f$ is a binary relation that associates for each element $a\in A'$ an element $f(a)$ of $B$.

More formally:

With $A' \subseteq A$, a relation $R\subseteq A \times B$ defines a function with domain $A'$ if every element $a$ of $A'$ is associated with exactly one element $b$ of $B$, and no element of $A\setminus A'$ is associated.

In other words: $\forall a\in A',\ \exists!b\in B\text{ such that }(a,b)\in R$, while no pair in $R$ has its first component in $A\setminus A'$.

If $A = A'$, then this function is an application.

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

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

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

Follow the moving current to read the direction of each edge.

Here $R = \{(X,1)\}$ is a function from $A$ to $B$.

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 $Y$ is a set of integers like $\{0,1\}$ 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.

Follow the moving current to read the direction of each edge.

Here $R = \{(X,1), (X,2)\}$ is a correspondence from $A$ to $B$. This is not a function because $X$ is associated with two elements of $B$. This is therefore not an application either.