Sets, Numbers and Probability Module

This first-year module on Sets, Numbers and Probability comprised of two parts and was taught by two different lecturers (with aliases L1, L2). The first part, taught in the Autumn Semester, is the Sets, Numbers and Proofs part. The information presented on the syllabus about the first part of the module is the following:

“The unit provides a thorough introduction to some systems of numbers commonly found in Mathematics: natural numbers, integers, rational numbers, modular arithmetic. It also introduces common set theoretic notation and terminology and a precise language in which to talk about functions. There is emphasis on precise definitions of concepts and careful proofs of results. Styles of mathematical proofs discussed include: proof by induction, direct proofs, proof by

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contradiction, contrapositive statements, equivalent statements and the role of examples and counterexamples.”

The mathematical areas that are covered in twenty lectures are Set Theory, with a focus on the notation used of sets; Venn diagrams, union and intersection, distributivity, the difference of 2 sets, complement, de Morgan’s laws; inclusion-exclusion principle and applications; power set and ordered pairs; cardinality and countability. The next topic presented are the basics of functions; injectivity and surjectivity of a function with examples of bijective functions and functions that are injective and not surjective and similarly surjective and not injective. Different approaches in proofs were also a part of this module, specifically direct proof; proof by induction; proof by contraction; and a discussion about examples and counterexamples. This part of the module also dealt with Number theory topics, namely Euclidean Algorithm; greatest common divisors; discussions about prime numbers; the fundamental theorem of arithmetic; rational and irrational numbers: irrationality of root 2; basics of modular arithmetic and equivalence relations. The other part of the module, taught in the Spring Semester, is Probability. The information provided on the syllabus of the course is given below:

“The term probability refers to the study of randomness and uncertainty. In any situation in which one of a number of possible outcomes may occur, the theory of probability provides methods for quantifying the chances or likelihood associated with the various outcomes. The study of probability as a branch of mathematics goes back over 300 Years and it is now a fundamental prerequisite for the study of statistics.” (bold in the original)

The areas, covered in this part of the module in eighteen lectures, are the following: Classical and modern definition of Probability; Kolmogorov’s axioms; basic properties proved from the Kolmogorov’s axioms; permutations; combinations; conditional probability; Binomial and Bayes’ theorem; independent events. The rest of the module focused on Discrete and Continuous samples. Specifically, after the presentation of the probability mass function and the cumulative distribution function; expectation and variance of different variables samples following the binomial, geometric, hypergeometric and Poisson distributions were

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presented. Similarly, in the Continuous samples after the discussion on the expectation and variance a presentation samples from uniform, Gaussian and exponential distributions followed. Additionally, there were some topics on Reliability and Markov Chains. However, these were not examined at the final examination at the end of the module, but reliability was examined in the coursework context.

The module was worth twenty credits, and it was compulsory for the first- year students. The distribution was forty percent from coursework and sixty percent of the grade from the final examination at the end of the year. The first twenty percent of the marks were given from two coursework sheets in the first semester from the Sets, Numbers and Proofs part of the course with questions focusing on sets and set operations; proof by induction; direct proof; and constructing examples; composition of functions; images of functions; injective, surjective and bijective functions; properties of divisors; reflexive; symmetric and transitive relations. The other twenty percent were given from a coursework sheet on Probability in the second semester. This coursework sheet had tasks on Kolmogorov’s axioms; the definition of disjoint events; combinations; probability mass and cumulative distribution function for discrete samples; proofs of relationships between expectation and variance of independent random variables; probability density and cumulative density functions of continuous random variables; Gaussian samples; Reliability functions; and parallel and series systems.

The students had access to a variety of materials. For the Sets, Numbers and Proofs part of the module the students were given six handouts, lecture notes covering the range of topics mentioned above; three exercise sheets and the solutions to those; two coursework sheets, their solutions and feedback on their solutions. They also had formative coursework and the solutions to that. For the Probability part of the module, the students had access to the old lecture notes, the new lecture notes, statistical tables, three exercise sheets, and three problem sheets and their solutions; a coursework sheet and the solutions produced by the lecturer.

The rest of the sixty percent of the grade was given from the final examination at the end of the academic year for both parts of the module. The examination had two compulsory and four optional tasks. Both parts of the module had one compulsory and two optional tasks. The examination lasted for two

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hours, and the students were asked to solve five tasks. These five tasks included both compulsory and three of the optional tasks. Each task was worth twenty marks and the pass grade of the examination is forty marks. The three tasks from the Sets, Numbers and Probability part of the module focused on: proof by induction; divisors; Euclidean Algorithm and Greatest Common Divisor; Unions and intersections of sets; reflexive, symmetric and transitive relations; injective and surjective functions; basics of modular arithmetic. The three tasks from the Probability part of the module were on Kolmogorov’s axioms and propositions following those axioms; probabilities of the union, intersection and conditional probability; Poisson random variable; expectation of discrete samples; expectation and variance of continuous random variables; probability density and cumulative distribution function and variables following the normal distribution.

4.2.2 Differential Equations and Applied Methods