- Lecturer: Ian Hewitt

Course Term: Michaelmas

Course Lecture Information: There will be 8 introductory lectures provided as videos.

Course Overview: The purpose of these introductory lectures is to establish some of the basic language and notation of university mathematics, and to introduce the elements of naïve set theory and the nature of formal proof.

Learning Outcomes:

Students should:

(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.

(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.

Course Synopsis:

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 \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}\), 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.

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.