## MA181-4-AU-CO:

Discrete Mathematics

## Key module for

BSC LG11 Economics and Mathematics,

BSC LG18 Economics and Mathematics (Including Foundation Year),

BSC LG1C Economics and Mathematics (Including Year Abroad),

BSC G100 Mathematics,

BSC G102 Mathematics (Including Year Abroad),

BSC G103 Mathematics (Including Placement Year),

BSC G104 Mathematics (Including Foundation Year),

MMATG198 Mathematics,

BSC 5B43 Statistics (Including Year Abroad),

BSC 9K12 Statistics,

BSC 9K13 Statistics (Including Placement Year),

BSC 9K18 Statistics (Including Foundation Year),

BSC G1G4 Mathematics with Computing (Including Year Abroad),

BSC G1G8 Mathematics with Computing (Including Foundation Year),

BSC G1GK Mathematics with Computing,

BSC G1IK Mathematics with Computing (Including Placement Year),

BSC G1F3 Mathematics with Physics,

BSC G1F4 Mathematics with Physics (Including Placement Year),

BSC G1F5 Mathematics with Physics (Including Foundation Year),

BSC GCF3 Mathematics with Physics (Including Year Abroad),

BSC I1G3 Data Science and Analytics,

BSC I1GB Data Science and Analytics (Including Placement Year),

BSC I1GC Data Science and Analytics (Including Year Abroad),

BSC I1GF Data Science and Analytics (Including Foundation Year),

MSCIG199 Mathematics and Data Science

## Module description

The first part of the module introduces the mathematics of sets in a non-axiomatic way, covering what is commonly referred to as naïve set theory. The versatility of using sets to define other mathematical objects is illustrated by studying functions and relations as sets. Further, the notions of countable and uncountable sets are explored.

Understanding and producing different types of mathematical proofs is an important part of the module. Besides standard techniques (direct proofs, proofs by contradiction, etc.) mathematical induction is introduced and studied as a powerful technique for proving statements about natural numbers.

The last part of the module introduces the basic ideas in propositional logic. This includes the use of truth tables, the laws of propositional logic, as well as the notion of a logical argument.

## Module aims

The aims of this module are:

- To provide a general understanding of sets and their connection to counting and defining other mathematical objects (including relations), mathematical proofs (especially inductive arguments), and the main ideas in propositional logic.

## Module learning outcomes

By the end of the module, students will be expected to:

- Have a basic knowledge of sets and the operations defined on them;
- Have a basic knowledge of binary relations and be able to check that a given relation is a partial order or an equivalence relation;
- Be able to compare the cardinalities of different sets using functions;
- Have a basic understanding of countable and uncountable sets;
- Be able to use mathematical induction;
- Have a basic understanding of propositional logic and be able to use truth tables for checking the validity of a logical argument.

## Module information

For students outside of the Department, an appropriate A level in Mathematics (or equivalent) is required for this module. If you are unsure whether you meet this criteria please contact maths@essex.ac.uk before attempting to enrol.

*Indicative syllabus:*

Sets:

Basic definitions

Set operations

Laws of set algebra

Principle of duality

Inclusion-exclusion for two sets

The power set of a set

Countable and uncountable sets

Relations:

Binary relations

Relation representations

Inverse of a relation

Composition of relations

Reflexivity, symmetry, transitivity, anti-symmetry

Computation of the transitive closure of a relation

Equivalence relations and equivalence classes

Partial and total orders

Functions:

Functions as relations

Composition of functions

Injective, surjective, and bijective functions

Inverse of a function

Induction and recursion:

Mathematical induction

Recursively defined sequences

Strong mathematical induction

Logic:

Propositions and logical connectives

Truth tables

Logical equivalence, tautologies, and contradictions

Laws of propositional logic

Logical arguments

## Learning and teaching methods

Teaching in the School will be delivered using a range of face to face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.

## Bibliography*

- Epp, S.S. (2019)
*Discrete mathematics with applications*. Fifth edition. Pacific Grove: Cengage. Available at: https://app.kortext.com/Shibboleth.sso/Login?entityID=https://idp0.essex.ac.uk/shibboleth&target=https://app.kortext.com/borrow/356308.

The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students.

Further reading can be obtained from this module's reading list.

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Assignment 1 | ||

Coursework | Assignment 2 | ||

Exam | Main exam: In-Person, Open Book (Restricted), 90 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 90 minutes during September (Reassessment Period) |

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

20% | 80% |

### Reassessment

Coursework | Exam |
---|---|

20% | 80% |

## External examiner

**50**hours,

**48 (96%)**hours available to students:

**2**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s), module, or event type.

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

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