intro to uni maths
Only two in-person lectures, which take place in the very first academic week. Lectured by Dr James Munro, whose resources for Oxford applicants have made him something of a celebrity for prospective students. I've been lucky enough to spend some time with him when I was working at an outreach programme; he is both a fantastic teacher whose enthusiasm for maths is infectious, and more generally a very lovely person.
The two problem sheets felt quite dense at the time, but looking back, they were a good introduction to what studying at Oxford is like.
syllabus
The natural numbers and their ordering. Induction as a method of proof, including a proof of the binomial theorem with non-negative integral coefficients.
Sets. Examples including , and intervals in . 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.