Mathematical Analysis I, 2004a

§§4–7. Relations. Mappings 15
(b) the set of all reals.
6. Prove that for any mapping f and any sets A, B, A i (i ∈ I),
(a) f −1 [A ∪ B] =f −1 [A] ∪ f −1 [B];
(b) f −1 [A ∩ B] =f −1 [A] ∩ f −1 [B];
(c) f −1 [A − B] =f −1 [A] − f −1 [B];
(d) f −1 [ ⋃ i A i]= ⋃ i f −1 [A i ];
(e) f −1 [ ⋂ i A i]= ⋂ i f −1 [A i ].
Compare with Problem 3.
[Hint: First verify that x ∈ f −1 [A] iffx ∈ D f and f(x) ∈ A.]
7. Let f be a map. Prove that
(a) f[f −1 [A]] ⊆ A;
(b) f[f −1 [A]] = A if A ⊆ D
f ′ ;
(c) if A ⊆ D f and f is one to one, A = f −1 [f[A]].
Is f[A] ∩ B ⊆ f[A ∩ f −1 [B]]?
8. Is R an equivalence relation on the set J of all integers, and, if so, what
are the R-classes, if
(a) R = {(x, y) | x − y is divisible by a fixed n};
(b) R = {(x, y) | x − y is odd };
(c) R = {(x, y) | x − y is a prime}.
(x, y, n denote integers.)
9. Is any relation in Problem 7 of §§1–3 reflexive? Symmetric? Transitive?
10. Show by examples that R may be
(a) reflexive and symmetric, without being transitive;
(b) reflexive and transitive without being symmetric.
Does symmetry plus transitivity imply reflexivity? Give a proof or
counterexample.
§8. Sequences ½
By an infinite sequence (briefly sequence) we mean a mapping (call it u) whose
domain is N (all natural numbers 1, 2, 3, ...); D u may also contain 0.
1 This section may be deferred until Chapter 2, §13.