MATH 242 – Principles of Analysis Problem Set 1 – due: Feb. 1

‘A‘ Section

1. Let A = {x ∈ R | x^{2} − 5x + 6 = 0}, B = (0, 4) = {x ∈ R | 0 < x < 4} and C = \{\frac{x}{x^{2}+1} | x ∈ R\} (Note: C is the range of the function f defined by f(x) = \frac{x}{x^{2}+1}.) a. Express the set C as a union of one or more closed intervals [a, b] in R. (Note: You should use facts from calculus to solve this. Don’t worry that we have not justified them yet.)

b. Find the sets A ∩ C and B ∩ C.

c. Find the sets B ∪ A and B ∪ C and express as unions of intervals in R.

2. Let B_{n} = {1, 1/4, 1/9, …, 1/n^{2}} for each natural number n ≥ 1. What are ∩_{n=1}^{∞}B_{n} and ∪_{n=1}^{∞}B_{n}.

3. Let I_{n} = [−1/n, 1/n] for any natural number n ≥ 1. What are ∩_{n=1}^{∞}I_{n} and ∪_{n=1}^{∞}I_{n}? (Explain your reasoning intuitively.)

4. Let f : R → R be the function defined by f(x) = x^{2} − 4x + 1.

a. Is f one-to-one? Why or why not?

b. Is f onto? Why or why not?

c. If I = (1, 3), what is the set f(I)? Explain. d. If J = (5, 6), what is the set f^{−1}(J). Explain.

‘B’ Section

1. Prove part (f) of Theorem 1.1.3 in the text. These are the De Morgan Laws for complements.

2. Let A and B be arbitrary sets. Does B = A−(A−B), as we might expect if we looked at the formula through the lens of ordinary algebra? If this is always true, prove it; if it is not, give both a counterexample (an example where the formula is not true), and a correct statement with proof.

3. Let f : A → B be a function.

a. Let C, D be subsets of A. Is it always true that f(C ∪ D) = f(C) ∪ f(D)? If this

is always true prove it; if it is not, give a counterexample. b. Show that f is onto if and only if f(f^{−1}(E)) = E for all subsets E of B.

4. Let f : A → B and g : B → C.

a. Show that if f and g are both one-to-one, then g ∘ f : A → C is also one-to-one. b. Is the converse of the statement in part a true? That is, if you know that g ∘

f is one-to-one, does it follow that f and g are one-to-one? Prove or find a counterexample.