286 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8
8.2 More on Functions and Mappings-
Surjections, Bijections,
Image, and Inverse Image
In the preceding article we saw that the definition of mapping (or, syn-
onymously, function from one set to another) provides for the specifica-
tion of a codomain, possibly distinct from the range (or image) of the asso-
ciated function. If f: A + B, then the general relationship between the
codomain B and rng f is rng f G B. In general, we say that f maps A into
B. In the special case, such as f: R + [0, oo), f(x) = x2, where rng f equals
B, we say that f maps A onto B. We formalize this condition in the following
definition.
DEFINITION 1
The mapping f: A -+ 6 is said to be onto, or a function that maps A onto 6, in
case rng f = B. We also say that such a mapping is surjective, or a surjection.
In view of the definition of rng f, we may characterize the onto property
as follows. The mapping f: A -+ B is onto if and only if, for every y E B,
there exists x E A such that f (x) = y [see Exercise l(a)]. Unlike the one-to-
one property, which depends only on the function part of a mapping, the
onto property depends in a crucial way on the choice of codomain. In fact,
given various mappings built on the same function, at most one of them can
be onto. As one example, iff: R + B, f (x) = sin x, where [ - 1, 11 G B s
R, f is onto if and only if B = [- 1, 11. In Exercise 2 you are asked to de-
termine whether or not various given mappings are surjective.
We suggested earlier that the injective and surjective properties of a
mapping are, in a sense, companion properties. This fact may not be evident
from a comparison of their definitions, but consider the following. A map-
ping f: A+ B is one to one if and only if each y E B has at most one x E A
such that f(x) = y, and onto if and only if each y E B has at least one x E A
such that f(x) = y. For mappings of R into R, the one-to-one property
/ dictates that every horizontal line meet the graph in at most one point; the
/
- / onto property requires that every horizontal line intersect the graph in
least one point. Recalling Theorem 4 of Article 8.1, we see another analogy
between the injective and surjective properties in the following result.
THEOREM 1
Let X be a nonempty set and let f: X + Y be a mapping. Then f is surjective if
and only if for any set Z and for any mappings g: Y -+ Z and h: y -+ Z such that
g~f=hof,wehaveg=h.
Proof * Assume f maps X onto Y, let Z be a set, and let g and h be
mappings of Y into Z such that g 0 f = h 0 f. To prove g = h, let y be
an arbitrary element of Y. We must prove that g(y) = h(y). Since f is
surjective, there exists x E X such that f (x) = y. Since g 0 f = h 0 f, by
