Map:
Representation of some features and
relations in some territory; in
mathematics:
function with specified domain and range. A.k.a. mapping.
The ideas of a map and the closely related mapping are very
fundamental, and are somehow involved in much or all of human cognition and
understanding  which after all is based on the making of mental maps or
models of things.
The first definition that is given is from the use of "map" in cartography
and the second from mathematics, but both are related, and mappings can be
seen as mathematical abstractions from maps.
1. maps: It is important to understand that one of the important points of maps
(that also applies to mappings) is that they leave out  abstract from, do not
depict  many things that are in the territory (or set) it represents. More
generally, the following points about maps are important:
· the map is usually not the territory (even if it is part
of it)
·
the map does usually not represent all of the territory but only
certain kinds of things occurring in the territory, in certain kinds of
relations
·
the map usually contains legenda and other instructions to
interpret it
·
the map usually contains a lot of what is effectively
interpunction
·
maps are on carriers (paper, screen, rock, sand)
·
the map embodies one of several different possible ways of
representing the things it does
·
the map usually is partial, incomplete and dated  and
· having
a map is usually
better
than
having
no map at all to understand the territory the map is about (supposing the map represents some truth)
·
maps may represent nonexisting territories and include guesses
and declarations to the effect "this is uncharted territory"
It may be well to add some brief comments and explanations to these points
Maps and territories: In the case of paper maps, the general point
of having a map is that it charts aspects of some territory (which can be seen
as a set of things with
properties in relations, but that is not
relevant in the present context).
Thus, generally a map only represents certain aspects of the territory it
charts, and usually contains helpful material on the map to assist a user to
relate it properly to what it charts.
And maps may be partially mistaken or may be outdated and still be helpful
to find one's way around the territory it charts, while it also is often
helpful if the map explicitly shows what is guessed or unknown in it.
2. mappings: In mathematics, the usage of the terms "map" and "function" is not
precisely regulated, but one useful way to relate them and keep them apart is
to stipulate that a function is a set of pairs
of which each first member is paired to just one second member, and a map is a
function of which also the sets from which the first and second members are
selected are specified. (These sets are known respectively as domain and
range, or source and target. See: Function.)
Note that for both functions and maps the rule or rules by which the first
members in the pairs in the functions and maps need not be known or, if it is
known, need not be explicitly given. Of course, if such a rule is known it may
be very useful and all that may need to be listed to describe the function or
map.
Here are some useful notations and definitions, that presume to some extent
standard set theory. It is assumed that the relations, functions and maps
spoken of are binary or twotermed (which is no principal restriction, since a
relation involving n terms can be seen as pair of n1 terms and the nterm).
In what follows "e" = "is a member of":
A relation R is a set of pairs.
A function f is a relation such that
(x)(y)(z)((x,y) e f & (x,z) e f > y=z).
A map m is a function f such that
(EA)(EB)(x)(y)((x,y) e f > xeA & yeB).
That m is a map from A to B is also written as:
"m : A > B" which is in words: "m maps A to B".
There are several ways in which such mappings can hold, and I state some
with the usual wordings:
m is a partial map of A to B:
m : A > B and not all xeA are mapped to some yeB.
m is a full map of A to B:
m is a map of A to B and not partial.
m is a map of A into B:
m : A > B and not all yeB are mapped to some xeA.
m is a map of A onto B:
m : A > B and not into.
One reason to have partial maps (and functions: the same terminology given
for maps holds for functions) is that there may well be exceptional cases for
some items in A. Thus, if m maps numbers to numbers using 1/n the case n=0
must be excluded.
