MATH10111: SETS, NUMBERS AND FUNCTIONS B (2011-2012)

Office hours, in room 2.214 of the Alan Turing Building: Thursdays 1:00-1:45; Fridays 11:15-12:00

Here is a list of definitions from weeks 1-5

Here is a list of definitions from weeks 7-12

Results from the course test in week 7 are now available listed by ID number here Scripts will be returned to you in Week 9, either by your supervisor, or if not then you can collect them from your advisor.

Course test question sheet

Course test answer sheet

Course test solution sheet

Course test feedback sheet

• Exercises for weeks 1-5
• Exercises for weeks 7-11
• Exercises for week 12

Solutions to exercises will appear on here (with an accouncement in the lecture) after you have had time to attempt the questions on your own.

• Solutions to exercise sheet 1
• Solutions to exercise sheet 2
• Solutions to exercise sheet 3
• Solutions to exercise sheet 4
• Solutions to exercise sheet 5
• Solutions to exercise sheet 7
• Solutions to exercise sheet 8
• Solutions to exercise sheet 9
• Solutions to exercise sheet 10
• Solutions to exercise sheet 11
• Solutions to exercise sheet 12

Syllabus

1. Introduction
2. Mathematical Logic. Propositions, predicates, logical connectives, truth tables. [3 lectures]
3. Proof by contradiction. Lots of examples. [2]
4. Induction proofs. Lots of examples. [4]
5. Set Theory. Sets, subsets, well known sets such as the integers, rational numbers, real numbers.Set Theoretic constructions such as unions, intersections, power sets, Cartesian products. [3]
6. Functions. Definition of functions,examples, injective and surjective functions, bijective functions, composition of functions, inverse functions. [3]
7. Counting. Counting of (mostly) finite sets, inclusion-exclusion principle, pigeonhole principle, binomial theorem. [3]
8. Euclidean Algorithm. Greatest common divisor,proof of the Euclidean Algorithm and some consequences, using the Algorithm. [3]
9. Congruence of Integers. Arithmetic properties of congruences,solving certain equations in integers. [3]
10. Relations. Examples of various relations,reflexive, symmetric and transitive relations. Equivalence relations and equivalence classes. Partitions. [3]
11. Some Number Theory. Fundamental theorem of Arithmetic, Fermat's little theorem. [2]
12. Binary Operations. Definition and examples of binary operations. Definition of groups and fields with examples. Proving that integers mod p ( p a prime) give a finite field. [4]

Textbooks

The course is based on the following text:

P.J. Eccles, An Introduction to Mathematical Reasoning: Numbers, Sets and Functions, Cambridge University Press, 1997.