Foundations of Pure Mathematics: Recent Episodes
The University of Nottingham
This module provides a foundation for all subsequent modules in Pure Mathematics. All of Pure Mathematics is written in the language of sets, functions and relations, and a large part of the module is devoted to gaining familiarity with both reading and writing this language.There will be an introduction to some basic counting principles, and the most important number systems will be introduced. Along the way, a variety of useful and interesting facts will be discussed. The module will include formal proofs and students will be given practice in writing proofs themselves. Topics to be covered will include: • counting problems, binomial coefficients; • the language of set theory; • relations and functions; • countable and uncountable sets. This module is aimed at first-year honours mathematics university students. However, pre-university students (including keen GCSE students) may find that much of the material is accessible. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area. See also Dr Feinstein's blog http://explainingmaths.wordpress.com/
Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module: discussions of Class Test 2.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The tenth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module Worksheet on functions and countability: one-sided inverses (left inverses and right inverses), injective, surjective and bijective functions, and countability.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure
The twentieth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers countability in terms of injections and surjections. The Cartesian product of two countable sets is countable. The set of rational numbers is countable. A countable union of countable sets is countable. Intervals of real numbers: the interval (0, 1) is uncountable (proved using Cantor’s diagonal argument), and it follows that every non-degenerate interval is uncountable.
These videos are also
The conclusion of the nineteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a brief discussion of sequences using up sets. Proof that the union of two countable sets is countable.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the Universit
The nineteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a brief discussion of different sizes of infinity. (The video “Beyond Infinity” may also be helpful here.) Sets with the same cardinality. Countable sets and uncountable sets defined. Connections between functions and sequences. Countability in terms of sequences. The union of two countable sets is countable. (Proved at the start of next class.)
These videos are also available on YouTube a
The ninth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet with a variety of questions: Cartesian products and set inclusions; Ssts, characteristic functions and congruence modulo 2; symmetric differences of sets.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Profe
The eighteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers disjoint permutations, especially disjoint cycles. Writing permutations as products of disjoint cycles.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The seventeenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers permutations of sets, especially finite sets. The two-line form for permutations. Inverses for permutations and the identity permutation. Multiplication of permutations using function composition.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr J
The eighth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet with a variety of questions: an application of the Binomial Theorem; working with sets and their characteristic functions, and the connection with the Inclusion-Exclusion Principle.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Fei
The sixteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers cardinality of finite sets: n-sets and k-subsets of sets. Discussion of injections, surjections and bijections between finite sets, including the Pigeonhole Principle. Contrast with infinite sets. Cardinalities of unions of sets: the Inclusion-Exclusion Principle (see Workshop 8 for more details). Binomial coefficients and the Binomial Theorem.
These videos are also available on YouTube at: https
The fifteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers graphs of functions revisited. Injections (injective functions), surjections (surjective functions) and bijections (bijective functions). The connection between bijections and inverse functions.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joe
The seventh workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet with a variety of questions: questions involving decimal expansions; modular arithmetic: squares and fourth powers modulo 5; finding an example of a function with specified properties.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com. Dr Joel
The fourteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers set-valued functions. Images and preimages (inverse images) of sets under functions: definitions, notation and example. Definition and examples of composition of functions. Associativity of function composition.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.word
The thirteenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers Informal definition of function, along with associated notation and terminology, including domain, codomain, argument, value, and notation. Definition of the graph of a function, with notation and examples. Formal definition of function (Bourbaki approach).
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's bl
The sixth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet with questions involving relations, equivalence relations, and modular arithmetic. In particular, looking at squares modulo 7, and applying this to show that a certain equation has no integer solutions.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpr
The twelfth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers notation for recurring decimal expansions. Real numbers with more than one decimal expansion. Real numbers have recurring/terminating decimal expansions if and only if they are rational.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein
The eleventh lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers the connection between equivalence relations, equivalence classes and partitions of sets. An introduction to modular arithmetic, especially modulo 2, modulo 9 and modulo 11.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associ
The fifth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers applications of the Fundamental Theorem of Arithmetic. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The tenth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers sets of subsets of a set. Unions of indexed collections of sets. Definition and examples of partitions of sets. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The ninth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers definitions and examples of equivalence relations, especially congruence modulo k. Brief discussion of equivalence classes (revisited in more detail in Lecture 11).. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Profes
The seventh lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers sets, subsets and set inclusions. Unions, intersections and differences of sets. De Morgan’s laws.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The fourth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a proof of Bézout’s Lemma. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The eighth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers definitions and examples of Cartesian products and (binary) relations, including order relations and congruence modulo k. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of N
The third workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet with questions about divisibility, prime numbers, and the square root of 3. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.
The sixth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers The statement of Bézout’s Lemma (see Workshop 4 for a proof). Application of Bézout’s Lemma to divisibility by primes. The statement of the Fundamental Theorem of Arithmetic (for integers n ≥ 2). These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Jo
The fifth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers: • The algebra of rational numbers: sums and products of rational numbers are rational (etc.) • Comparison with irrational numbers: investigation of combinations of rational numbers and irrational numbers. • Density of the rationals and the irrationals in the real line. Note the factor 2 error in the last sketch proof, as noted in the call-out.
These videos are also available on You
The second workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet based on working with definitions. Questions involving sums of subsets of the real line, set inclusions and set equalities. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the Un
The fourth lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers Highest Common Factor (HCF, also known as Greatest Common Divisor, GCD) for integers, and lowest terms for rational numbers. How and why indirect proof (proof by contradiction) works. Examples of indirect proofs, including the proof that the square root of two is irrational.
These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Fe
The third lecture in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers an introduction to formal definitions and direct proofs, with discussion, motivation and examples. The dangers of backwards logic. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the Univers
The first workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module is a detailed look at some extracts from the Module Information Document including: Module aims and learning outcomes. The importance of understanding and being able to demonstrate. understanding of the material. Likely style of exam questions. The importance of looking at the feedback on student performances in the last two years' exams. What you need to do if you want to do well in this module, rather t
The second class in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers Sets and their elements. Number systems. Elementary reasoning using examples and counterexamples.Various ways to specify sets, and simplifying their descriptions. Intervals and number line sketches. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Fei
Motivation for pure mathematical thinking (definitions, proofs and examples) illustrated by looking at some introductory puzzles, with voting. These videos are also available on YouTube at: https://www.youtube.com/playlist?list=PLpRE0Zu_k-BzsKBqQ-HEqD6WVLIHSNuXa Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham.