Quick info
Instructor: Alex Youcis
Email: alex.youcis@utoronto.ca (see syllabus for email policy!)
Office hours: Mon 2:30–4 (PGB003)
Syllabus: PDF
MAT138 — Introduction to Proofs (Fall 2025)
| Lecture — Date | Topic / Notes | Text (chapter) | Board photos |
|---|---|---|---|
| Lecture 1 — Sep 3, 2025 (Wed) | Anatomy of a proof, proof that \(\sqrt{2}\) is irrational | Chapters 1–2 | Lecture 1 photos |
| Lecture 2 — Sep 8, 2025 (Mon) | Definition of primes, some basic properties, and infinitude of primes | Chapter 3 | Lecture 2 photos |
| Lecture 3 — Sep 10, 2025 (Wed) | Precisely stated the fundamental theorem of arithmetic, proved existence portion, defined function \(v_p\) and proved basic properties about it | Chapter 3 + supplemental | Lecture 3 photos |
| Lecture 4 — Sep 15, 2025 (Mon) | Discussed Euclidean algorithm, proved Bezout's lemma, stated Euclid's proposition, and deduced uniqueness in fundamental theorem of arithmetic | Chapter 3 + supplemental | Lecture 4 photos |
| Lecture 5 — Sep 17, 2025 (Wed) | Gave proof of Euclid's proposition, talked about weak and strong mathematical induction, did many examples | Chapter 4 | Lecture 5 photos |
| Lecture 6 — Sep 22, 2025 (Mon) | Went through various proofs exploiting symmetry | Chapter 5 | Lecture 6 photos |
| Lecture 7 — Sep 24, 2025 (Wed) | Discussed several combinatorial proofs, noticably involving permutations and the function \(n\choose k)\) | Chapter 5 | Lecture 7 photos |
| Lecture 8 — Sep 29, 2025 (Mon) | We discussed many examples of proofs without words, how they can help, and how they can lie. | Chapter 6 | Lecture 8 photos |
| Lecture 9 — Oct 1, 2025 (Wed) | We discussed the (sub)set operations of union, intersection, difference, symmetric difference, complement, and Cartesian product. | Supplemental | Lecture 9 photos |
| Lecture 10 — Oct 6, 2025 (Mon) | We discussed relations, gave several examples, and talked aboout the properties of symmetric, reflexive, and transitive. | Chapter 11 | Lecture 10 photos |
| Lecture 11 — Oct 8, 2025 (Wed) | We discussed further properties of relations, notably talking about reflexive/symmetric/transitive closures of relations and defining equivalence relations. | Chapter 11 | Lecture 11 photos |
| — Oct 13, 2025 (Mon) | No lecture (holiday) | — | — |
| Lecture 12 — Oct 15, 2025 (Wed) | We discussed equivalence relations, partitions, and the beginning of functions. | Chapter 11 | Lecture 12 photos |
| — Oct 20, 2025 (Mon) | Midterm review | — | — |
| — Oct 22, 2025 (Wed) | Midterm | — | — |
| — Oct 27, 2025 (Mon) | Reading week — no lecture | — | — |
| — Oct 29, 2025 (Wed) | Reading week — no lecture | — | — |
| Lecture 13 — Nov 3, 2025 (Mon) | We discussed injections, bijections, and surjections as well as inverse functions. | Chapters 11 and 13 | Lecture 13 photos |
| Lecture 14 — Nov 5, 2025 (Wed) | We discussed the notion of cardinality, gave many examples, and defined the notion of (un)countability. | Chapters 11 and 13 | Lecture 14 photos |
| Lecture 15 — Nov 10, 2025 (Mon) | We gave a cleaner description of Cantor's proof of the uncountability of \(\mathbb{R}\), proved that \(\aleph_0\) is the smallest infinite cardinal, proved that subtracting a finite set from an infinite one doesn't change cardinality, and proved Cantor's theorem that for any set \(X\) one has \(\# 2^X>\# X\). | Chapter 11 and 13 | Lecture 15 photos |
| Lecture 16 — Nov 12, 2025 (Wed) | We gave an alternative presentation of Cantor's proof that \(\# 2^X>\# X\), discussed its relation to the previous one via the natural bijection betwen \(\mathcal{P}(X)\) and \(2^X\), proved that \(\# \mathbb{R}=2^{\aleph_0}\), discussed the Contiuum Hypothesis, defined algebraic and transcendental numbers, discussed why the transcendentalness of \(\pi\) was so significant in the context of squaring the circle, and finally proved that the algebraic numbers are countable and the transcendental numbers are uncountable. | Chapters 11 and 13 | Lecture 16 photos |
| Lecture 17 — Nov 17, 2025 (Mon) | We started discussing the notion of partially ordered sets (posets), gave some examples, and defined various types of maps (e.g., monotone maps and isomorphisms). | Chapter 14 | Lecture 17 photos |
| Lecture 18 — Nov 19, 2025 (Wed) | We discussed further examples of posets and isomorphism, explained why the notion of isomorphic allowed us to identify ostensibly different posets and how this extra structure can help distinguish isomorphic sets (e.g., \(\mathbb{N}\) and \(\mathbb{Q}\)), we then stated Cantor's theory characterizing \((\mathbb{Q},\leqslant)\). | Chapter 14 | Lecture 18 photos |
| Lecture 19 — Nov 24, 2025 (Mon) | TBA | TBA | To be posted |
| Lecture 20 — Nov 26, 2025 (Wed) | TBA | TBA | To be posted |