Attempt QUESTIONS 1 and 2, and THREE other questions. Do not turn over until you are told to do so by the Invigilator.

Transcription

1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination SET THEORY MTHE6003B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHE6003B Module Contact: Dr David Asperó, MTH Copyright of the University of East Anglia Version: 1

2 - 2 - Note: Any theorem you use must be clearly stated. A theorem can be used without proof unless you are required to prove it. Assume the base theory ZF unless otherwise specified. 1. (i) Define the notion of transitive set and the notion of ordinal. [8 marks] (ii) Is every nonzero ordinal a successor ordinal? Justify your answer. (iii) Is every nonzero ordinal a limit ordinal? Justify your answer. 2. (i) State the Cantor Bernstein Schröder Theorem. (ii) Define the notion of cardinal. (iii) Is every ordinal a cardinal? Justify your answer. [8 marks] MTHE6003B Version: 1

3 (i) Prove that if n ω and f : n n is an injective function, then f is a bijection between. Conclude from this that every natural number is a cardinal. (ii) Prove that if X and Y are isomorphic transitive sets, then X = Y and the identity on X is the unique isomorphism f : (X, ) (Y, ). [7 marks] (iii) Let X and Y be sets and let and be well founded partial orders on, respectively, X and Y. Can there be more than one order isomorphism between (X, ) and (Y, )? Justify your answer. [7 marks] 4. (i) Working in ZFC, prove that if R is a relation on a set X and a X, then there is a set Y X satisfying the following properties. (a) a Y. (b) Y ℵ 0. (c) For every b Y, if there is some c X such that brc, then there is some d Y such that brd. (ii) (a) State the Axiom of Choice. (b) Prove that the following statements are equivalent modulo ZF. The Axiom of Choice. For every two sets X and Y, at least one of the following holds. There is an injective function i : X Y. There is an injective function i : Y X. MTHE6003B PLEASE TURN OVER Version: 1

4 Let Φ : <ω ω ω be a bijection and let A = {{Φ(b n) : n ω} b : ω 2} (i) Prove that A is not a mad family. [8 marks] (ii) Working in ZFC, prove that there are at least 2 ℵ 0 many mad families M such that A M. [12 marks] 6. (i) Provethatforeveryuncountable cardinal κ thereisapartitionof κ into Reg κ many stationary sets, where Reg denotes the class of infinite regular cardinals. (ii) Let us say that a stationary subset S of a cardinal κ reflects if and only if there is some ordinal α < κ such that S α is a stationary subset of α. Prove that if κ is an infinite cardinal such that every stationary subset of κ + reflects, then κ has to be singular. END OF PAPER MTHE6003B Version: 1

5 MTHE6003B (Set Theory) exam feedback (main series 2016/17) There was a mix of responses to the six questions on the exam. Many of the low marks in the exam corresponded to people who seem to have had confused ideas about some of the more theoretical notions seen in the module. Questions 1 and 2 were meant to be very easy, and in fact most people obtained very high marks here, especially for Question 1. Questions 3 and 4 were more challenging. Finally, Questions 5 and 6 were meant to be the most difficult. However, the average mark for Question 5 is slightly higher than that for Question 4. Question 6 was attempted (very successfully) by only one student. 1