Foundations of Computer Science

Learning Outcomes

Knowledge and Understanding

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

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

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
• Calculate probabilities of events and recognise probability distributions
• Critically analyse and solve counting problems on finite, discrete structures
• Apply operations on vectors and matrices and solve systems of linear equations
• Recognise, understand and construct rigorous mathematical proofs
• Use statistical analysis, including sampling, hypothesis testing and regression

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.

• Probability and statistics
• Introduction to probability: elementary probability formulae, discrete and continuous probability distributions.
• Introduction to statistics: sampling, confidence intervals, hypothesis testing, regression.

Algebra

• Linear and quadratic equations, systems of equations.
• Polynomials: basic properties and operations.
• Vectors: basic properties, scalar product, vector product.
• Matrix algebra: basic properties, inverse, determinant, Eigenvalues, Eigenvectors,
• Solving systems of linear equations.

Summative

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

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.