8.2 MORE ON FUNCTIONS AND MAPPINGS 273Exercises
- Use Definition 1 and the definition of rng f (Definition 4, Article 7.1) to prove
that a mapping f: A -+ B is a surjection if and only if, for each y E B, there exists
x E A such that y = f(x). - Determine whether each of the following mappings is (i) a surjection, (ii) an in-
jection, (iii) a bijection: - (a) Use the definition of onto mapping to prove that iff maps X onto Y and
g maps Y onto 2, then g 0 f maps X into Z [(b) of Theorem 21.
(b) In each of the following three cases, give examples of two mappings f: A -, B
and g: B -+ C such that:
(i) g^0 f: A -, C is a surjection, but f is not surjective. (What must be true of
g in any such example?)
(ii) g is surjective, but g^0 f is not surjective. (What must be true off in any
such example?)
(iii) f is surjective, but g 0 f is not surjective. (What must be true of g in any
such example?)
*(c) Prove that iff: X -+ Y is surjective and g; Y -, 2, then g^0 f: X -, Z is a sur-
jection if and only if g is surjective.
(d) state and prove a result analogous to (c) for injective mappings. - (a) Prove the corollary to Theorem 2. That is, prove that if f: X -+ Y and
g: Y -, Z so that go f:X -, Z, then:
(i) Iff and g are bijections, then g^0 f is a bijection.
(ii) If g 0 f is a bijection, then f is injective and g is surjective.
(b) In each of the following three cases, give example of mappings: f: X -, Y and
g: Y - Z such that:
(i) g^0 f: X -+ Z is a bijection, but f is not onto and g is not one to one
(ii) f is bijective, but g^0 f is not bijective
(iii) g is bijective, but g 0 f is not bijective - (a) Use the result of Theorem 1 to give a proof of (b) of Theorem 2, different
from the proof in Exercise 3(a). [Recall the proof of (c) Theorem 2, given in the
text.]
(6) Use Theorem 1 to give a proof of (d) of Theorem 2, different from the proof
given in the text.