MATH 242 – Principles of Analysis Problem Set 1 – due: Jan. 31

‘A‘ Section

1. Assume that A, B are sets of integers.

a. What is the contrapositive of the statement: “If x is even then x ∈ A ∪ B”?

Express without using not. b. What is the converse of the statement in part a?

2. Let A = {x ∈ R | x^{2} − 5x + 4 = 0}, B = (0, 1) = {x ∈ R | 0 < x < 1} and C = \{\frac{x}{x^{2}+9} | x ∈ R\} (Note: C is the range of the function f defined by f(x) = \frac{x}{x^{2}+9}.) a. Express the set C as a closed interval [a, b] in R or as a union of such intervals. (Note: You should use facts from calculus to solve this. Don’t worry that we have not justified them yet.)

b. Find the sets B ∩ A and B ∩ C.

c. Find the sets A ∪ B and A ∪ C and express using set notation.

3. For n a general natural number, let B_{n} = {0, 2n}. What are ∩_{n=1}^{∞}B_{n} and ∪_{n=1}^{∞}B_{n}?

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

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

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

b. Is f onto? Why or why not?

c. If I = (0, \sqrt{3}), what is the set f(I)? Explain.

d. If J = (−π/4, π/4), 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 DeMorgan Laws for unions and intersections.

2. Let A and B be arbitrary sets. Does A = A−(B −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).

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 one-to-one if and only if f^{−1}(f(C)) = C for all subsets C of A.

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

onto, does it follow that f and g are onto? Prove or find a counterexample.