# Mapping/Introduction/Section

## Definition

Let
and
denote sets. A *mapping* from to is given by assigning, to every element of the set , exactly one element of the set . The unique element which is assigned to
,
is denoted by . For the mapping as a whole, we write

If a mapping
is given, then is called the *domain* (or domain of definition) of the map and is called the *codomain* (or *target range*) of the map. For an element
,
the element

is called the *value* of at the *place* (or *argument*) .

Two mappings
and
are equal if and only if their domains coincide, their codomains coincide and if for all
the equality
in
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 .

For every set , the mapping

which sends every element to itself, is called the *identity* (on ). We denote it by
. For another set and a fixed element
,
the mapping

which sends every element
to the *constant value* , is called the *constant mapping*
(with value ).
It is usually again denoted by .^{[1]}

There are several ways to describe a mapping, like value table, bar chart, pie chart, arrow diagram, the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule, which allows computing the values of the mapping for every argument. Such rules are e.g.
(from to )
, , etc. In the sciences and in sociology also *empirical functions* are important which describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximated) in mathematical manner.

- ↑ Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things.