4 Introduction
Fig. 0.
A mapping with rule f, domain G, and range contained in P(Y) is
denoted f: G ---t P(Y), which is read "the mapping of G into P(Y) by f."
Also, in a loose way, we write f : X ---t Y and read "the mapping from X
into Y by f" or "the mapping from X to Y by f."
The notation f : G ---t P(Y) emphasizes the fact that the idea of a set
mapping involves three objects: The rule f and two sets G and P(Y). Thus
the mapping f : G ---t P(Y) and f : QI ---t P(Y), where G' C P(X) and
G' -=j:. G, are considered as different, although the rule f and the set P(Y)
are the same in both cases.The mappings f : G ---t P(Y) and g : G ---t P(Y) are defined to be
equal if, and only if, f(A) = g(A) for every A E G. In this case we write
f = g, it being understood that the domain and codomain are the same
in both instances.
When reading "the mapping of G into P(Y) by f," the term "into" is
used to convey the idea that the range of the mapping need not fill up P(Y).
If it does happen that the range of the mapping is P(Y), the mapping is
called onto or surjective, denoted by f : G onto> P(Y), or by f : G =t P(Y),which is read "the mapping of G onto P(Y) by f."
Now suppose that G 1 = { { x} : x EX}, that is, that G1 is a class of thesingleton sets of P(X), and that f assigns to each { x} E G 1 a singleton
{y} E P(Y). Then the mapping is called a single-valued function with
domain G1 and range contained in P(Y). Since { x} E G1 and x E X
are equivalent relations (in the sense that each relation implies the other),
and similarly, {y} E P(Y) is equivalent to y E Y, it is usually said that
the function f has domain X and range contained in Y, and we write f :
X ---t Y. The element {y} is called the image of { x} under f, and also the
value off at { x }, and we usually write y = f( x) instead of {y} = f( {x} ).
The notation x ---t f(x) is also employed to indicate such a mapping. For
instance, we write x ---t sin x, the domain and codomain of the function