Discrete Mathematics
This is a basic course for undergraduate students.Organization
For the Winter Semester 2025-2026, lectures are scheduled on Thursdays at 14:00 in room S9 at Mala Strana.Information about course requirements (Exam etc)
The final grade will depend on your performance in the exam at the end of the semester. To be able to take the exam you need to obtain a "pass" in the tutorials. Details can be found here.
Syllabus
Check SIS to see a tentative list of topics to be covered. The exact material covered during the lectures will be updated on this webpage as the course progresses.
Following is a list of Books and other material relevant to the lectures.
Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil Notes on Probability by Jiří Matoušek
Material Covered in the Lectures
October 02: Basics of logic; Proof: (example) square of an odd number is odd; Direct proof: (example) There exists irrational numbers \(p,q\) such that \(p^q\) is rational; Proof by contradiction: (example) \(\sqrt{2}\) is irrational; Well-defined sets; writing a set; operations on sets: union, intersection, difference, complement; Basic notation \(\mathbb{N,Z,Q,R}\); Mathematical induction; Inductive proof of \(\sum_{i=1}^{n}i=\frac{n\cdot(n+1)}{2}\); Fibonacci numbers; Proof by strong induction, example \(F_n=\frac{\phi^n-\psi^n}{\sqrt{5}}\).
Recommended reading: Sections 1.1-1.3
October 09: Notation \(2^X\); number of subsets of a set by induction; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Equivalence and orderings (definition).
Recommended reading: Sections 1.3, 1.5
October 16: Equivalence relation: partition into equivalence classes; Orderings; linear/total vs. partial; Poset; minimal, minimum, maximum, maximum elements; Every finite poset has at least one minimal element; Drawing posets; Hasse diagram; chains and anti-chains.
Recommended reading: Section 1.6, Chapter 2
October 23: Large implies tall or wide; Application Erös-Szekeres theorem; functions: injection, surjection, bijection; Number of functions from an \(n\)-element set to an \(m\)-element set; number of injective functions functions from \([n]\) to \([m]\).
Recommended reading: Sections 2.4, 3.1
October 30: number of bijections; permutations: two-line and one-line notation; number of permutations of a finite set; Binomial coefficients; notation \( {n\choose k}, {X\choose k}\); double counting; \(\left|{X\choose k}\right|={|X|\choose k}\); \(\sum_{k=0}^n{n\choose k}=2^n\); Pascal's identity; Binomial theorem (proof by induction); applications: \(\sum_{k=0}^n {n\choose k}=2^n\), number of even and odd cardinality subsets.
Recommended reading: Sections 3.1-3.3
November 06: Number of nonnegative integer solutions of \(x_1+x_2+\cdots+x_r=n\); Estimates of \(n!\); Estimates of \(\binom{n}{k}\); Inclusion-Exclusion principle.
Recommended reading: Sections 3.3-3.7
November 13: Application of inclusion-exclusion principle: Number of derangements; Graphs; Examples: \(K_n, K_{n,m}\); Graph isomorphism; Number of non-isomorphic graphs; More examples of graphs: cycle, path; Handshake lemma; Subgraphs/Induced subgraphs.
Recommended reading: Sections 3.8, 4.1-4.2
November 20 (Plan): connectedness; walk; (closed) tour; Eulerian graphs; A graph is Eulerian if and only if it is connected and every vertex has even degree; Graph score, score theorem.
Recommended reading: Sections 4.2-4.4
| Invitation to Discrete Mathematics | by Jiří Matoušek and Jaroslav Nešetřil |
| Notes on Probability | by Jiří Matoušek |
Material Covered in the Lectures
October 02: Basics of logic; Proof: (example) square of an odd number is odd; Direct proof: (example) There exists irrational numbers \(p,q\) such that \(p^q\) is rational; Proof by contradiction: (example) \(\sqrt{2}\) is irrational; Well-defined sets; writing a set; operations on sets: union, intersection, difference, complement; Basic notation \(\mathbb{N,Z,Q,R}\); Mathematical induction; Inductive proof of \(\sum_{i=1}^{n}i=\frac{n\cdot(n+1)}{2}\); Fibonacci numbers; Proof by strong induction, example \(F_n=\frac{\phi^n-\psi^n}{\sqrt{5}}\).
Recommended reading: Sections 1.1-1.3
October 09: Notation \(2^X\); number of subsets of a set by induction; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Equivalence and orderings (definition).
Recommended reading: Sections 1.3, 1.5
October 16: Equivalence relation: partition into equivalence classes; Orderings; linear/total vs. partial; Poset; minimal, minimum, maximum, maximum elements; Every finite poset has at least one minimal element; Drawing posets; Hasse diagram; chains and anti-chains.
Recommended reading: Section 1.6, Chapter 2
October 23: Large implies tall or wide; Application Erös-Szekeres theorem; functions: injection, surjection, bijection; Number of functions from an \(n\)-element set to an \(m\)-element set; number of injective functions functions from \([n]\) to \([m]\).
Recommended reading: Sections 2.4, 3.1
October 30: number of bijections; permutations: two-line and one-line notation; number of permutations of a finite set; Binomial coefficients; notation \( {n\choose k}, {X\choose k}\); double counting; \(\left|{X\choose k}\right|={|X|\choose k}\); \(\sum_{k=0}^n{n\choose k}=2^n\); Pascal's identity; Binomial theorem (proof by induction); applications: \(\sum_{k=0}^n {n\choose k}=2^n\), number of even and odd cardinality subsets.
Recommended reading: Sections 3.1-3.3
November 06: Number of nonnegative integer solutions of \(x_1+x_2+\cdots+x_r=n\); Estimates of \(n!\); Estimates of \(\binom{n}{k}\); Inclusion-Exclusion principle.
Recommended reading: Sections 3.3-3.7
November 13: Application of inclusion-exclusion principle: Number of derangements; Graphs; Examples: \(K_n, K_{n,m}\); Graph isomorphism; Number of non-isomorphic graphs; More examples of graphs: cycle, path; Handshake lemma; Subgraphs/Induced subgraphs.
Recommended reading: Sections 3.8, 4.1-4.2
November 20 (Plan): connectedness; walk; (closed) tour; Eulerian graphs; A graph is Eulerian if and only if it is connected and every vertex has even degree; Graph score, score theorem.
Recommended reading: Sections 4.2-4.4