Foundations of Pure Mathematics

Video Lectures

Displaying all 29 video lectures.
Lecture 1
Introduction to Pure Mathematics
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Introduction to Pure Mathematics
The first class in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers motivation for pure mathematical thinking (definitions, proofs and examples) illustrated by looking at some introductory puzzles with voting.
These videos are also available for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 2
Sets and Numbers
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Sets and Numbers
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 3
Workshop 1: About this module
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Workshop 1: About this module
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 than just passing it.
These videos are also available for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 4
Definitions and Direct Proofs
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Definitions and Direct Proofs
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 5
Rational and Irrational Numbers
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Rational and Irrational Numbers
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 6
Workshop 2
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Workshop 2
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 7
More on Rational and Irrational Numbers
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More on Rational and Irrational Numbers
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 8
Bezout's Lemma and Prime Factorization
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Bezout's Lemma and Prime Factorization
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 9
Workshop 3
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Workshop 3
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 10
Sets and Subsets
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Sets and Subsets
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 11
Cartesian Products and Relations
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Cartesian Products and Relations
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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..
These videos are also available for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 12
Workshop 4
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Workshop 4
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 13
Equivalence Relations and Equivalence Classes
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Equivalence Relations and Equivalence Classes
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 14
Unions and Partitions
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Unions and Partitions
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 15
Workshop 5
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Workshop 5
The fifth workshop in Dr Joel Feinstein's G11FPM Foundations of Pure Mathematics module covers a worksheet based on applications of the Fundamental Theorem of Arithmetic. Counting the number of different positive factors (divisors) of a positive integer. Divisibility. Highest powers of primes dividing a given positive integer. Another way to prove that certain real numbers are irrational..
These videos are also available for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 16
Equivalence Classes and Modular Arithmetic
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Equivalence Classes and Modular Arithmetic
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem... 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.
Lecture 17
Decimal expansions and rational numbers
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Decimal expansions and rational numbers
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 18
Workshop 6
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Workshop 6
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 19
Functions and their graphs
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Functions and their graphs
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 20
Functions and sets
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Functions and sets
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.

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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 21
Workshop 7
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Workshop 7
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 22
Properties of functions
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Properties of functions
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 23
Finite sets and cardinality
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Finite sets and cardinality
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 24
Workshop 8
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Workshop 8
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com.
Lecture 25
Permutations of finite sets
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Permutations of finite sets
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 26
Permutations continued
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Permutations continued
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 27
Workshop 9
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Workshop 9
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
Dr Feinstein's blog may be viewed at: http://explainingmaths.wordpress.com.
Lecture 28
Cardinality for infinite sets
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Cardinality for infinite sets
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.
Lecture 29
Conclusion of Cardinality for infinite sets
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Conclusion of Cardinality for infinite sets
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 for download on iTunes U at: https://itunes.apple.com/us/itunes-u/foundations-pure-mathem...
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.