## MATH10111 - Sets, Numbers and Functions B

**Year:**1 -

**Semester:**1 -

**Credit Rating:**15

#### Requisites

Corequisites | |

MATH10131 Calculus and Vectors B |

#### Aims

TBA

#### Brief Description

This lecture course is intended to introduce students to the concept of "proof". The objects of study, sets, numbers and functions, are basic to almost all Mathematics.

#### Learning Outcomes

On successful completion of this module students will be

- familiar with and able to manipulate the basic concepts of Pure Mathematics such as sets and functions;
- able to construct elementary proofs of mathematical statements utilizing inductive arguments and arguments by contradiction;
- able to understand proofs of such results at the Fundamental Theorem of Arithmetic and the Euclidean Algorithm;
- familiar with the definitions and know some examples of groups and fields.

#### Syllabus

- Mathematical Logic. Propositions, predicates, logical connectives, truth tables. [3 lectures]
- Proof by contradiction. Lots of examples.
- Induction proofs. Lots of examples.
- 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.
- Functions. Definition of functions,examples, injective and surjective functions, bijective functions, composition of functions, inverse functions.
- Counting. Counting of (mostly) finite sets, inclusion-exclusion principle, pigeonhole principle, binomial theorem.
- Euclidean Algorithm. Greatest common divisor,proof of the Euclidean Algorithm and some consequences, using the Algorithm.
- Congruence of Integers. Arithmetic properties of congruences,solving certain equations in integers.
- Relations. Examples of various relations,reflexive, symmetric and transitive relations. Equivalence relations and equivalence classes. Partitions.
- Some Number Theory. Fundamental theorem of Arithmetic, Fermat's little theorem.
- 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.

#### Teaching & Learning Process (Hours Allocated To)

Lectures | Tutorials/Example Classes | Practical Work/Laboratory | Private Study | Total |
---|---|---|---|---|

33 | 11 | 0 | 106 | 150 |

#### Assessment and Feedback

- Supervision attendance and participation; Weighting within unit 10%
- Coursework; In class test, weighting within unit 15%
- Two and a half hours end of semester examination; Weighting within unit 75%

#### Further Reading

The course is based on the following text:

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

#### Staff Involved

Dr Charles Eaton - Lecturer

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