256 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8
it is. Furthermore, with this understanding in force, we regard two func-
tions as equal if and only if they contain the same ordered pairs. The
result in Exercise 2(a) follows directly from this criterion.
There are mathematical contexts in which we do care about a cartesian
product containing a function f as a subset. We approach this situation
in the following paragraphs by defining the notion of a function from a set
A to a set B, otherwise known as a mapping.MAPPINGS
DEFINITION 2
A function from a set A to a set 8, denoted f: A -+ B (also known as mapping
from A to 8, or simply a mapping) consists of a function f, satisfying dom f = A,
and a set B such that rng f c B. The set B is called the codomain of f.The notation f: A -* B from Definition 2 is usually read "f is a mapping
from the set A to the set B," or "f is a function that maps the set A to the
set B." In view of the discussion preceding Definition 2, the concept of
"function from A to B or "mapping (from A to B)" differs in a subtle way
from that of function; a mapping consists of a function plus something
more. In order to define a mapping, we must specify along with the func-
tion f (whose definition already determines the sets dom f and rng f) a
set B such that rng f r B. As an example, we might consider a constant
mapping of R into R, given by C: R -, R, where C(x) = c, c E R; the codo-
main R contains the range (c), as required by the definition of mapping.
Or, a mapping could be defined by g: R -* [I, co), where g(x) = cosh x. In
this case the codomain [I, co) not only contains the range of g as a subset,
but in fact equals rng g. Most particularly, it is possible to have two dis-
tinct mappings, both of which involve the same function. The mappings
k: R -+ R and k: R + [0, a), where k(x) = x4, illustrate this situation. We
say that mappings f: A -+ B and g: C -+ D are equal if and only iff = g
(which implies automatically that A = C) B = D.
The notion of mapping may be used to amplify concepts encountered
earlier in the text. As one example, an infinite collection of sets (A,, A,,.. .}
indexed by N (recall Article 4.2) may be viewed as a mapping of N into
any set containing each of the sets in the collection as an element, such
as P(U,"=, Ai). More generally, any infinite sequence is a mapping of N
into some codomain (such as R or C).ONETO-ONE MAPPINGS
At various earlier stages of the text (and surely at several points in your
prior mathematical training), the notion of a one-to-one function has come
into play (see, in particular, Exercise 6, Article 5.2). There are various ways
of characterizing "one-to-oneness," in addition to the definition "f(x,) =