Mathematics I

Learning Outcomes

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Elementary ideas of probability theory
• Basic counting techniques and their applications to common data structures
• The language of set theory and common operations on sets, including infinite sets
• Principles of mathematical proof and sound logical reasoning
• Logical systems and the concept of formal proof
• The interplay of syntax and semantics in mathematics, logic and computer science
• Functions and relations as fundamental structures in computer science

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Use the language of logic and set theory in order to make precise formal statements
• Recognise, understand and construct rigorous mathematical proofs
• Critically analyse and solve counting problems on finite, discrete structures
• Calculate probabilities of events and recognise discrete probability distributions

Mathematical proof

• Proof by case analysis, proof by contradiction.
• Induction and recursion.
• Universal properties.

Sets, functions and relations

• Basic notation, representations and examples. Membership and subsets.
• Operations on sets: union, sum, intersection and complement.
• Pairs, tuples, cartesian products, powersets.
• Relations, equivalence relations and partial orders.
• Functions: injections, surjections, bijections.
• Cardinality, infinite sets.

Logic

• Propositional logic. Logical connectives.
• Syntax and semantics.
• Natural deduction, soundness and completeness.
• Quantifiers and predicate logic.

Combinatorics

• Basic principles of counting: product and sum rules, inclusion-exclusion principle, pigeonhole principle.
• Combinations, permutations and arrangements, binomial theorem.

Introduction to trees and graphs: directed, undirected and weighted.

Introduction to probability

• Elementary probability formulae, discrete probability distributions

The content of this module is delivered through lectures, the module website, directed reading, pre-recorded materials and tutorials.

Students work on their understanding through a combination of independent study, preparation for timetabled activities, tutorials, and problem classes, along with formative assessments in the form of problem sheets.

Summative

This is how we’ll formally assess what you have learned in this module.