Answers to Exercises

Summary

1.1. The elements of E are precisely the points not in C(E); so are the elements of C(C(E)).

1.2. (a) Every letter is either a consonant or a vowel, and all the vowels occur in “real functions.”

(b) C(E) consists of all the vowels.

(c) C(F) consists only of consonants (in fact, not all of them).

(d) F ∩ E = {r, l, f, n, c, t, s}, which contains no vowels.

1.3. Not very practical. (Repeat the preceding discussion with “bibliography” replacing “set.”)

2.1. (a) All numbers greater than or equal to 1; all nonpositive numbers; 1; 0.

(b), (c), (d), (e): The same as (a).

(f) All nonnegative numbers; all nonpositive numbers; 0; 0.

(g), (h) No upper bound; all nonpositive numbers; +∞; 0.

(i) All numbers greater than or equal to 5π/6; all numbers less than or equal to π/6; 5π/6; π/6.

2.2. Every nonempty set E that is bounded below has a greatest lower bound, denoted by inf E, with the properties that every x ∈ E satisfies x ≥ inf E, and if A > inf E there is at least one x ∈ E such that x < A. If the least upper bound property is assumed and E is a set that is bounded below, let F consist of all numbers x such that –x ∈ E. Since x > M means –x < –M, the set F is bounded above and so has a least upper bound B.