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mappings

Mappings are a foundational concept describing how elements from one set, called the domain, are related to elements of another set, called the codomain. In mathematics, a mapping is typically a function: a rule that assigns to every input in the domain exactly one output in the codomain. When a rule is defined only for some inputs, it is a partial mapping; when it covers all inputs, it is a total mapping.

A mapping is often denoted by f: A → B, where A is the domain and B is

Mappings have several important properties. An injective (one-to-one) mapping assigns distinct outputs to distinct inputs; a

In computer science, a mapping often refers to a data structure that associates keys with values, such

Mappings also appear in cartography and geographic information systems, where a cartographic mapping represents real-world features

Overall, mappings provide a formal way to relate elements across sets, with wide applications across mathematics,

the
codomain.
The
image
f(A)
is
the
set
of
outputs
that
actually
occur,
a
subset
of
B.
For
a
given
y
in
B,
the
preimage
f^{-1}(y)
is
the
set
of
inputs
that
map
to
y.
The
graph
of
a
mapping
is
the
set
of
ordered
pairs
(x,
f(x)).
surjective
(onto)
mapping
covers
the
entire
codomain;
a
bijective
mapping
is
both
injective
and
surjective
and
thus
establishes
a
one-to-one
correspondence
between
domain
and
codomain.
If
a
mapping
is
bijective,
it
has
an
inverse
mapping
that
reverses
the
assignment.
as
dictionaries
or
maps.
Operations
include
lookup,
insertion,
and
deletion,
with
performance
typically
tied
to
the
underlying
implementation
(for
example,
hash
maps
or
balanced
trees).
using
coordinate
systems
and
projections
to
translate
space
onto
a
planar
surface.
computer
science,
and
geography.