Discrete Mathematics
This is a basic course for undergraduate students.Organization
For the Winter Semester 2026-2027, lectures are scheduled on Mondays at 15:40 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.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
Entries for lectures that have not taken place yet are shown dimmed and marked as plan; they contain tentative material and will be updated as the course progresses.
October 05:plan
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
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 12:plan
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
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 19:plan
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
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 26:plan
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 from \([n]\) to \([m]\).
Recommended reading: Sections 2.4, 3.1
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 from \([n]\) to \([m]\).
Recommended reading: Sections 2.4, 3.1
November 02:plan
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
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 09:plan
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; Application of inclusion-exclusion principle: number of derangements.
Recommended reading: Sections 3.3-3.8
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; Application of inclusion-exclusion principle: number of derangements.
Recommended reading: Sections 3.3-3.8
November 16:plan
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; 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 4.1-4.4
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; 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 4.1-4.4
November 23:plan
Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations; Spanning tree of a graph.
Recommended reading: Sections 5.1, 5.3
Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations; Spanning tree of a graph.
Recommended reading: Sections 5.1, 5.3
November 30:plan
Planar drawing; planar graph; faces of a planar drawing; Euler formula for planar graphs; number of edges in planar graphs; number of edges in triangle-free planar graphs; Application: \(K_5, K_{3,3}\) are not planar; Graph subdivision; Kuratowski's theorem (without proof): planar iff no subgraph that is a subdivision of \(K_5, K_{3,3}\).
Recommended reading: Sections 6.1-6.3
Planar drawing; planar graph; faces of a planar drawing; Euler formula for planar graphs; number of edges in planar graphs; number of edges in triangle-free planar graphs; Application: \(K_5, K_{3,3}\) are not planar; Graph subdivision; Kuratowski's theorem (without proof): planar iff no subgraph that is a subdivision of \(K_5, K_{3,3}\).
Recommended reading: Sections 6.1-6.3
December 07:plan
Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability; Independent events; Finite version of Boole's inequality: \(P\left[\bigcup_{i=1}^n B_i\right]\leqslant\sum_{i=1}^n P[B_i]\), equality when \(B_i\)'s are disjoint; Application: Bayes' theorem.
Recommended reading: Section 10.2
Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability; Independent events; Finite version of Boole's inequality: \(P\left[\bigcup_{i=1}^n B_i\right]\leqslant\sum_{i=1}^n P[B_i]\), equality when \(B_i\)'s are disjoint; Application: Bayes' theorem.
Recommended reading: Section 10.2
December 14:plan
Probability that a random graph is bipartite; Random variable, expectation; Indicator function; Linearity of expectation; Examples: expected number of heads in a sequence of \(n\) tosses of a fair coin; Every graph has a balanced bipartite subgraph with at least half the number of edges.
Recommended reading: Sections 10.3-10.4
Probability that a random graph is bipartite; Random variable, expectation; Indicator function; Linearity of expectation; Examples: expected number of heads in a sequence of \(n\) tosses of a fair coin; Every graph has a balanced bipartite subgraph with at least half the number of edges.
Recommended reading: Sections 10.3-10.4
January 04:plan
Platonic solids and planar graphs; Proof that only five kinds of platonic solids exist; Coloring maps: the four-color theorem (without proof); Definition: Dual graph; Chromatic number problem; \(d\)-degenerate graphs; \(\chi(G)\leqslant d+1\) if \(G\) is \(d\)-degenerate; \(\chi(G)\leqslant 6\) if \(G\) is planar; \(\chi(G)\leqslant 5\) if \(G\) is planar.
Recommended reading: Sections 6.3, 6.4
Platonic solids and planar graphs; Proof that only five kinds of platonic solids exist; Coloring maps: the four-color theorem (without proof); Definition: Dual graph; Chromatic number problem; \(d\)-degenerate graphs; \(\chi(G)\leqslant d+1\) if \(G\) is \(d\)-degenerate; \(\chi(G)\leqslant 6\) if \(G\) is planar; \(\chi(G)\leqslant 5\) if \(G\) is planar.
Recommended reading: Sections 6.3, 6.4
End of lectures.