Module Directory

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.

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.

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

1. Have a basic knowledge of sets and the operations defined on them;


2. Have a basic knowledge of binary relations and be able to check that a given relation is a partial order or an equivalence relation;


3. Be able to compare the cardinalities of different sets using functions;


4. Have a basic understanding of countable and uncountable sets;


5. Be able to use mathematical induction and strong mathematical induction;


6. Have a basic understanding of propositional logic and be able to use truth tables for checking the validity of a logical argument.

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

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.