Introduction to University Mathematics (2022-23)
Main content blocks
- Lecturer: Profile: Ian Hewitt
(i) have the ability to describe, manipulate, and prove results about sets and functions using standard mathematical notation;
(ii) know and be able to use simple relations;
(iii) develop sound reasoning skills;
(iv) have the ability to follow and to construct simple proofs, including proofs by mathematical induction (including strong induction, minimal counterexample) and proofs by contradiction;
(v) learn how to write clear and rigorous mathematics.
Sets. Examples including \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}\), and intervals in \(\mathbb{R}\). Inclusion, union, intersection, power set, ordered pairs and cartesian product of sets. Relations. Definition of an equivalence relation. Examples.
Functions: composition, restriction; injective (one-to-one), surjective (onto) and invertible functions; images and preimages.
Writing mathematics. The language of mathematical reasoning; quantifiers: "for all", "there exists". Formulation of mathematical statements with examples.
Proofs and refutations: standard techniques for constructing proofs; counter-examples. Example of proof by contradiction and more on proof by induction.
Problem-solving in mathematics: experimentation, conjecture, confirmation, followed by explaining the solution precisely.
Section outline
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The primary course materials are these pdf lecture notes and the Panopto lecture videos (easiest to access via the direct links to the right, under Panopto/Completed recordings, once the site has recognised you as having access).
There are 16 videos, mostly around 20-25 mins. I recommend watching at x1.25 speed. The notes and videos contain more or less the same material, occasionally in a slightly different order. The rough correspondence is:
Lecture 1 - The natural numbers and induction (Sections 1.1,1.2)
Lecture 2 - The binomial theorem and introduction to sets (Sections 1.3,2.1)
Lecture 3 - Algebra of sets, cardinality (Sections 2.2,2.3,2.4)
Lecture 4 - Logical notation, relations, and equivalence relations (Sections 5.1,3.1,3.2,3.3)
Lecture 5 - Functions (Sections 4.1,4.2,4.3)
Lecture 6 - Handling logical notation and quantifiers (Sections 5.1,5.2)
Lecture 7 - Constructing mathematical statements and proofs (Sections 5.3,6.1,6.2)
Lecture 8 - Problem solving examples (Section 6.3)
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Covers lectures 1-4. The following would be a sensible order to do things:
Lectures 1 and 2. Questions 1,2
Lecture 3. Questions 3,4
Lecture 4. Questions 5,6,7
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Covers lectures 5-8. The following would be a sensible order to do things:
Lecture 5. Questions 1,2
Lecture 6. Questions 3,4,5
Lectures 7 and 8. Questions 6,7,8
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NB: If you have difficulty accessing these quizzes or the videos, it may be that you need to 'enrol' in this course, see here.
The quizzes contain short questions to recap ideas from each lecture and to check your understanding. I recommend taking a few minutes to do them as you work through the videos or notes.
Some of these should be quite straightforward, while others require careful thought. Getting them wrong doesn't matter at all - you may even learn more by getting them wrong. (You can try the questions as many times as you like - don't be put off by things the system says about grading - these don't count for anything!)
Click 'Attempt quiz now' to start. Click 'Check' to check your answers, 'Try another question like this one' to have another go (it really just means 'Try again' but I can't change what it says), and 'Next Page' to move to the next question.
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