Mapping/Introduction/Section

Definition

Let ${}L$ and ${}M$ denote sets. A mapping ${}F$ from ${}L$ to ${}M$ is given by assigning to every element of the set ${}L$ exactly one element of the set ${}M$. The unique element that is assigned to ${}x\in L$ is denoted by ${}F(x)$. For the mapping as a whole, we write

$F\colon L\longrightarrow M,x\longmapsto F(x).$

If a mapping ${}F\colon L\rightarrow M$ is given, then ${}L$ is called the domain (or domain of definition) of the map, and ${}M$ is called the codomain (or target range) of the map. For an element ${}x\in L$, the element

${}F(x)\in M\,$

is called the value of ${}F$ at the place (or argument) ${}x$.

Two mappings ${}F\colon L_{1}\rightarrow M_{1}$ and ${}G\colon L_{2}\rightarrow M_{2}$ are equal if and only if their domains coincide, their codomains coincide, and if for all ${}x\in L_{1}=L_{2}$ the equality ${}F(x)=G(x)$ in ${}M_{1}=M_{2}$ holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called functions. However, we will usually reserve the term function for mappings where the codomain is a number set like the real numbers ${}\mathbb {R}$.

For every set ${}L$, the mapping

$L\longrightarrow L,x\longmapsto x,$

which sends every element to itself, is called the identity (on ${}L$). We denote it by ${}\operatorname {Id} _{L}$. For another set ${}M$ and a fixed element ${}c\in M$, the mapping

$L\longrightarrow M,x\longmapsto c$

that sends every element ${}x\in L$ to the constant value ${}c$ is called the constant mapping (with value ${}c$). It is usually again denoted by ${}c$.

There are several ways to describe a mapping, like a value table, a bar chart, a pie chart, an arrow diagram, or the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule that allows computing the values of the mapping for every argument. Such rules are, e.g., (from ${}\mathbb {R}$ to ${}\mathbb {R}$) ${}x\mapsto x^{2}$, ${}x\mapsto x^{3}-e^{x}+\sin x$, etc. In the sciences and in sociology, also empirical functions are important that describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximately) in a mathematical manner.