8.1 FUNCTIONS AND MAPPINGS 255Figure 8.1 Graph of the "cubing function."the type described in Exercise 3(b). It should be noted also that the identity
function and constant functions may be considered in much wider contexts
than that of the real numbers as domain and range. If A is any set, the
identity function on A, denoted I,, is defined by the rule I,(x) = x for all
x E A. If A is any set and c E A, the constant function C is defined by
C(x) = c for all x E A.
The alert reader may have noticed an inconsistency between our discus-
sion thus far of the function concept and the definition of relation, given
earlier. If, as Definition 1 states, a function f is a particular type of re-
lation, then according to Definition 2, Article 7.1, there must be an un-
derlying cartesian product A x B containing f as a subset, where we
necessarily have dom f E A and rng f E B. Technically, this is true, with
the result that, strictly speaking, we should not be able to define a function
simply by giving a rule (or even by listing a set of ordered pairs), but rather,
should also have to specify sets A and B. This situation has the rather dis-
turbing consequence that the result in Exercise 2(a) becomes false. A set of
ordered pairs satisfying the condition in Definition 1, possibly determined
by a given domain and rule of correspondence, could correspond to many
different functions, as we vary the sets A and B. It is a fact of mathe-
matical life thaf we often wish to specify a function simply by giving a
domain and rule or a rule with the domain understood, or by describing
explicitly a set of ordered pairs, and do not wish to be entangled in a
cartesian product. Indeed, we have proceeded this way in Examples 1 and
- Let us agree, then, that we may view a function as being properly defined
by the various methods exhibited in Examples 1 and 2, even though the
underlying cartesian product may not be identified explicitly. When this is
done, the understanding is that an appropriate cartesian product exists but
is irrelevant to our current application, so that in essence we don't care what