Name: Student Id: There are nine problems (check that you have 9 pages). Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Write your answers and proofs in the space provided. Good luck! Some notation: N, Q and R denote the set of natural numbers, rational numbers and real numbers respectively. For a pair of sets A and B, B A denotes the set of all functions from A to B. 1
1. Provide examples. Explanations are not necessary. Describe a bijection between the open interval (0, 1) and R. Describe a bijection between the closed interval (0, 1) and the half-open interval (0, 1]. Describe a bijection between 2 N and P(N); here 2 = {0, 1}. Provide an example of a subset S P(N) such that S and P(N)\S are uncountable. [Bonus: Justify your answer.] 2
Proposition 1 Let f : X Y and g : Y X be functions. Suppose that g f = I X. Then (a) f is injective, and (b) g is surjective. 2. For the Proposition above: Prove conclusion (a). Provide example of a pair of functions f and g illustrating the Proposition for which f is not a bijection. (No explanations required.) 3
Definition 2 (Countable Sets) A set A is countable if either A is finite or there is a bijection N A. If a set is not countable, the it is called uncountable. Theorem 3 Let A be a set. The following statements are equivalent. A is countable. There is an injective function A N. There is a surjective function N A. 3. Let A and B sets. Prove the following statements. If A is countable and f : A B is a surjective function, then B is countable. If A is uncountable and f : A B is an injective function, then B is uncountable. 4
Theorem 4 (Cantor s First Diagonal Argument.) If A 0, A 1, A 2,... is a sequence of countable sets, then n=0 A n = A 1 A 2 A 3... is a countable set. 4. Prove that if A and B are countable sets, then A B = {(a, b): a A, b B} is a countable set. 5. Prove that the set of polynomials p(x) = a 0 + a 1 x + + a n x n with integer coefficients is a countable set. 5
Theorem 5 (Cantor s Second Diagonal Argument.) The set of infinite binary sequences is an uncoutable set. 6. Describe the proof of the above result, i.e., prove that the set 2 N is uncountable. 6
7. A counting problem. State Schröder-Bernstein Theorem. A binary relation on a set A is a subset R A A. The relation R is symmetric if for all a, b A, (a, b) R implies that (b, a) R. Prove that the set all symmetric binary relations on N has the same cardinality as 2 N. 7
8. A function f : N N is strictly increasing iff m, n N (m < n f(m) < f(n)). Prove that the set of strictly increasing functions f : N N is uncountable. 8
[Blank page] 9
MATH 3300 Test 2 Solutions are expected to be short. In the case of proofs, one or two short paragraphs should be the average length. Good luck! 1. (15 points) Provide precise statements for (a), (b), (c), (d). Provide examples for (e), (f), (g), no proofs are required. (a) Schröder-Bernstein Theorem. (b) Axiom of Choice. (c) Well ordering principle. (d) The Continuum Hypothesis. (e) Well ordered set (W, ) with infinitely many elements with no immediate predecessor. (f) A set of cardinality (ℵ 3 0 + 1)(2 3ℵ 0 + ℵ ℵ 0 1 ). (g) A bijection f : N N Z (a graphical description is ok); What is f(7) for your function? 2. (15 points; Axiom of Choice) Recall that X Y if and only if there is a one-to-one function X Y. Prove the following statements; point out any use of the Axiom of choice. (a) If f : A B is onto, then there is h: B A such that f h is the identity function on B. (b) If f : A B is surjective, then B A. (c) Without using the axiom of choice, prove that if f : N B is surjective, then B N. 3. (15 points; Cardinal Arithmetic) (a) Prove that n ℵ 0 = ℵ 0 for every n N with n 1. (b) Let x, y, z be cardinals. Prove that (x y ) z = x yz. (c) Prove that 2 ℵ 0 = n ℵ 0 for every n N with n > 1. 4. (5 points; First Uncountable Ordinal.) Prove that if A is a countable subset of S Ω, then A has an upper bound in S Ω. 5. (5 points; Well Ordering Principle.) Prove that the well-ordering principle implies the axiom of choice. 6. (15 points; Exponentiation of Well Orders.) Let (A, A ) and (B, B ) be well ordered sets. For f, g B A define f g if and only if there exists a A such that f(a) < B g(a) and f(x) = g(x) for every x S a. Prove the following statements. (a) is a transitive relation on B A. (b) is an irreflexive relation on B A. (c) For every pair f, g B A either f = g, f g, or g f. (d) ** (Not part of the test) Every non-empty subset of (B A, ) has a smallest element. 1