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.
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.