KMB, 7/10/99

M1F Foundations of Analysis—Problem Sheet 1

This is assessed coursework. Please hand in solutions to the starred questions (i.e., to questions 1–4) at the end of the lecture on Friday 15th October.

1^{∗}) (5 marks) Let A be the set {1, 2, 3, 4, 5}. Let B be the set {1, 2, 3, {4}, 5}. Let ℤ denote the set of all integers. Which of the following statements are true and which are false? (Just write T or F as required.)

\begin{matrix}(a) 1 ∈ A & (f) A ⊆ B \\ (b) \{1\} ∈ A & (g) B ⊆ A \\ (c) \{1\} ⊆ A & (h) ℤ ∈ A \\ (d) A = B & (i) A ∈ ℤ \\ (e) A ∈ B & (j) A ⊆ ℤ \end{matrix}

2^{∗}) (3 marks) Let A be the set {x ∈ ℝ : 0 < x < 3}. Let B be the set {x ∈ ℝ : 0 < x < 2}. Let C be the set {x ∈ ℤ : 0 < x < 2}. Let D be the set {1}. Which of these four sets contains B? Which contains C? Are any of these sets equal? Briefly explain your answers.

3^{∗}) (6 marks) For each of the following statements, decide whether they are true or false. For the true statements, write down a proof. For the false ones, write down a counterexample.

(a) For all real numbers x, we have x^{2} > 0.

(b) If a and b are irrational numbers, then so is a + b.

(c) If n is an integer, then n^{2} + n is always even.

4^{∗}) (6 marks) Let S denote the set of real numbers less than 1. As in Q3, decide whether the following statements are true, and give either a proof or a counterexample as required.

(a) If x ∈ S and y ∉ S then x < y.

(b) If x ∈ S then there exists y ∈ S with x < y.

(c) If x ∉ S then there exists y ∉ S with x > y.

5) Let S be the set of rational numbers x such that x^{2} < 2. Prove that if x ∈ S then there exists y ∈ S such that x < y. Hint: if x = a/b ∈ S with a, b > 0 then you may wish to consider the rational number y = \frac{3a+4b}{2a+3b}.

6) If a and b are integers such that 3a + b and 4a + 2b are both multiples of 2000, then prove that a and b are both multiples of 1000. Must a and b both be both multiples of 2000?