274 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8
(c) Use the result of Theorem 4, Article 8.1, to give a proof of (a) of Theorem 2,
different from the proof in Exercise lqb), Article 8.1.
- (a) Prove parts (a) and (b) of Theorem 3: that is, prove that, given sets A and B:
(i) The identity mapping I,: A -, A on A is a bijection.
(ii) Iff: A 4 B is a bijection, then f -': B -, A is a bijection.(b) Prove that iff: A -, B and g: B -* C are bijections, then:
(i) (g^0 f)- ' is a bijection from C to A
(ii) f - '^0 g- ' is a bijection from C to A
(iij) (go f)-'=f log-'(c) According to the text (see the paragraph after the proof of Theorem 3), two
sets A and B are numerically equivalent if and only if there exists a bijection
from one to the other. Use previously proved results in this chapter to prove
that the relation of numerical equivalence is an equivalence relation on the col-
lection of all subsets of any given universal set U.
- (a) Given f: R -+ R, f(x) = 3x - 7, calculate:
(c) Given f: R -* R, f(x) = (x + I),, calculate:
- (a) Prove parts (b) through (e) of Theorem 4; that is, iff: A -, B, then:
0) f - '(%I = % (ii) A= f-'(~)
(iii) rng f = f(A)
(iv) If MI, M, E A and M, G M,, then f(M,) G f(M,)(b) (i) Give an example of a mapping f: A -, B and subsets MI, and M, of A
such that MI c M,, but f(M,) = f(M,).
(ii) Prove that iff: A -, B is one to one, then for any subsets MI and M, of
A, if f(M,) = f(M,), we may conclude MI = M,.
(c) Prove (d) of Theorem 5; that is, given a mapping f: A -, B, f (f - '(N)) =
N for each subset N of B if and only iff is an onto mapping.
- (a) Prove (a) of Theorem 6; that is, iff: A - B, then:
(I) f-'(N, u N,)= f-'(NJu f-'(N,)foranysubsets N, and N,ofB
(ii) f - '(N, n N,) = f - '(N ,) n f - '(N,) for any subsets N, and N, of B