Discrete Mathematics
This is a basic course for undergraduate students.Organization
For the Winter Semester 2024-2025, lectures are scheduled on Mondays at 10:40 in room S9 at Mala Strana. Tutorials are held on Tuesdays at 10:40 and at 12:20 in Troja.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
September 30: 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 07: Notation \(2^X\); number of subsets of a set by induction; functions: injection, surjection, bijection; number of subsets ofa set via bijection with set of binary strings; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Equivalence and orderings (definition).
Recommended reading: Sections 1.4-1.6
October 14: Number of functions from an \(n\)-element set to an \(m\)-element set; number of injective functions functions from \([n]\) to \([m]\); 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; number of nonnegative integer solutions of \(x_1+x_2+\ldots+x_r=n\).
Recommended reading: Sections 3.1-3.3
October 21: Binomial theorem (proof by induction); applications: \(\sum_{k=0}^n {n\choose k}=2^n\), number of even and odd cardinality subsets; Equivalence relation: partitition into equivalence classes; Orderings; linear/total vs. partial; Poset; minimal, maximal, minimum, maximum elements; Every finite poset has at least one minimal element; Drawing posets: Hasse diagram; chains and anti-chains; Notation: \(\alpha(P),\omega(P)\); Theorem: \(\alpha(P)\omega(P)\geqslant |P|\);
Recommended reading: Sections 3.3, 1.6, 2.1-2.4
November 04: Theorem: \(\alpha(P)\omega(P)\geqslant |P|\); Application: Erdös-Szekeres theorem; Estimates of \(n!\); Inclusion-Exclusion principle (without proof); Application: Number of derangements.
Recommended reading: Sections 2.4, 3.5-3.8
November 11: Estimate of \(\binom{n}{k}\); Inclusion-Exclusion principle (proof); Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability; Independent events
Recommended reading: Sections 3.6, 3.7, 10.2; Notes on Probability by Jiří Matoušek
November 18: 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; Probability that a randomly picked graph on \(n\) vertices 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; Graphs; Examples: \(K_n, K_{n,m}\); Every graph has a balanced bipartitite subgraph with at least half the number of edges (proof using expectation).
Recommended reading: Sections 10.2-10.4; Notes on Probability by Jiří Matoušek
November 25: 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.
Recommended reading: Sections 4.1, 4.2, 4.4
December 02: Graph score, score theorem; Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations.
Recommended reading: Sections 4.3, 5.1
December 09: 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 minors; Graph subdivision; Kuratowski's theorem (without proof): planar iff no subgraph that is a subdivision of \(K_5, K_{3,3}\); Wagner's theorem (without proof): planar iff no \(K_5, K_{3,3}\) minor.
Recommended reading: Sections 6.1-6.3
December 16: 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
| Invitation to Discrete Mathematics | by Jiří Matoušek and Jaroslav Nešetřil |
| Notes on Probability | by Jiří Matoušek |
Material Covered in the Lectures
September 30: 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 07: Notation \(2^X\); number of subsets of a set by induction; functions: injection, surjection, bijection; number of subsets ofa set via bijection with set of binary strings; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Equivalence and orderings (definition).
Recommended reading: Sections 1.4-1.6
October 14: Number of functions from an \(n\)-element set to an \(m\)-element set; number of injective functions functions from \([n]\) to \([m]\); 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; number of nonnegative integer solutions of \(x_1+x_2+\ldots+x_r=n\).
Recommended reading: Sections 3.1-3.3
October 21: Binomial theorem (proof by induction); applications: \(\sum_{k=0}^n {n\choose k}=2^n\), number of even and odd cardinality subsets; Equivalence relation: partitition into equivalence classes; Orderings; linear/total vs. partial; Poset; minimal, maximal, minimum, maximum elements; Every finite poset has at least one minimal element; Drawing posets: Hasse diagram; chains and anti-chains; Notation: \(\alpha(P),\omega(P)\); Theorem: \(\alpha(P)\omega(P)\geqslant |P|\);
Recommended reading: Sections 3.3, 1.6, 2.1-2.4
November 04: Theorem: \(\alpha(P)\omega(P)\geqslant |P|\); Application: Erdös-Szekeres theorem; Estimates of \(n!\); Inclusion-Exclusion principle (without proof); Application: Number of derangements.
Recommended reading: Sections 2.4, 3.5-3.8
November 11: Estimate of \(\binom{n}{k}\); Inclusion-Exclusion principle (proof); Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability; Independent events
Recommended reading: Sections 3.6, 3.7, 10.2; Notes on Probability
November 18: 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; Probability that a randomly picked graph on \(n\) vertices 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; Graphs; Examples: \(K_n, K_{n,m}\); Every graph has a balanced bipartitite subgraph with at least half the number of edges (proof using expectation).
Recommended reading: Sections 10.2-10.4; Notes on Probability
November 25: 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.
Recommended reading: Sections 4.1, 4.2, 4.4
December 02: Graph score, score theorem; Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations.
Recommended reading: Sections 4.3, 5.1
December 09: 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 minors; Graph subdivision; Kuratowski's theorem (without proof): planar iff no subgraph that is a subdivision of \(K_5, K_{3,3}\); Wagner's theorem (without proof): planar iff no \(K_5, K_{3,3}\) minor.
Recommended reading: Sections 6.1-6.3
December 16: 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