Numbers and Sets - exercises for enthusiasts 2. W. T. G.

1. Does there exist an uncountable family B of subsets of ℕ such that for every A, B ∈ B (distinct) the intersection of A with B is finite?

2. Is it possible to write the closed interval [0, 1] as a countably infinite union of disjoint closed (non-empty) subintervals?

3. Let f be a function from ℝ^{2} to ℝ such that, for every x and y in ℝ, the functions z ↦ f(x, z) and w ↦ f(w, y) are polynomials. Prove that f is a polynomial in x and y. Does the result hold if ℝ is replaced by ℚ?

4. Let X be a non-empty set with an associative binary operation defined on it (which is written as multiplication in what follows). Suppose that for every x ∈ X there is a unique x^{∗} such that xx^{∗}x = x. Prove that X is a group.

5. Is there a field that can be made into an ordered field in exactly three ways?

6. Given n points in the plane, not all collinear, show that it is possible to find a line containing exactly two of them.

7. Let G and H be groups such that G × ℤ ≡ H × ℤ. Show that G and H need not be isomorphic, but that they must if they are Abelian.

8. If X is a set, let X^{(r)} denote the set of subsets of X of size r. Show that if ℕ^{(r)} is coloured with finitely many colours, then there is an infinite subset X ⊂ ℕ such that all the sets in X^{(r)} have the same colour.

9. Call a set X of positive integers a multiplicative basis of order 2 if every positive integer n can be written as xy with x and y in X. Suppose that X is a multiplicative basis of order 2. Prove that there is a positive integer m that can be written as xy in at least 2003 different ways, all with x and y in X.

10. Let ..., a_{−1}, a_{0}, a_{1}, a_{2}, … be a doubly infinite sequence of points in [0, 1] such that for every i, a_{i+1} is either a_{i} + \sqrt{2} (mod 1) or a_{i} + \sqrt{3} (mod 1). Can the set A = \{a_{i} : i ∈ ℤ\} be closed? (That means that any point not in A is contained in an open interval disjoint from A.)

11. Given a real number x, write log^{2} x for log log x, log^{3} x for log log log x and so on. Define a function f on real numbers greater than or equal to 1 by f(x) = x(log x)(log^{2} x) … (log^{k} x), where k is the largest integer for which log^{k} x is greater than or equal to 1. (If k = 0 then f(x) is interpreted to be x.) Determine whether the sum ∑_{n=1}^{∞} \frac{1}{f(n)} converges or diverges.