Discrete Mathematics
This is a basic course for undergraduate students.Organization
For the Winter Semester 2023-2024, the lectures are scheduled on Mondays at 14:00 in room S5 at Mala Strana. Tutorials are held on Wednesdays 12:20 in S11, Wednesdays 14:00 in S10, and Thursdays 9:00 in S11.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; Basic notation \(\mathbb{N,Z,Q,R}\); Mathematical induction.
Recommended reading: Sections 1.1-1.3
October 09: Well-defined sets; writing a set; operations on sets: union, intersection, difference, complement; functions: injection, surjection, bijection; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Number of subsets of a set.
Recommended reading: Sections 1.4-1.6
October 16: Notation \(2^X\); number of subsets of a set via bijection with set binary strings; number of functions from an \(m\)-element set to an \(n\)-element set; number of injective functions functions from \([m]\) to \([n]\); 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).
Recommended reading: Sections 3.1-3.3
October 23: Inclusion-Exclusion principle (proof by double counting); Application: number of surjective functions. Orderings; linear/total vs. partial; Poset; minimal, maximal, minimum, maximum elements; Every finite poset has at least one minimal element.
Recommended reading: Sections 3.7, 2.1-2.2
October 30: Drawing posets: Hasse diagram; Chains, antichatins; Notation: \(\alpha(P),\omega(P)\); Theorem: \(\alpha(P)\omega(P)\geqslant |P|\); Application: Erdös-Szekeres theorem; Counting the number of nonnegative integer solutions of \(x_1+x_2+\ldots+x_r=n, l_i \leqslant x_i \leqslant u_i\); Equivalence relation: equivalence classes, partition of a set.
Recommended reading: Sections 2.3--2.4, 3.3, 1.6
November 06: Equivalence relation: partitition into equivalence classes; Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability.
Recommended reading: Sections 1.6, 10.2; Notes on Probability by Jiří Matoušek
November 13: 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; Indpendendent events; 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 bipartitite subgraph with at least half the number of edges (proof outline using expectation).
Recommended reading: Sections 10.2--10.4; Notes on Probability by Jiří Matoušek
November 20: Every graph has a bipartitite subgraph with at least half the number of edges (proof using expectation); Graph isomorphism; More examples of graphs: cycle, path, Handshake lemma; Subgraphs/Induced subgraphs; connectedness; walk; circuit; Eulerian graphs; A graph is Eulerian if and only if every vertex has even degree.
Recommended reading: Sections 4.1, 4.2, 4.4, 10.4
November 27: A graph is Eulerian if and only if every vertex has even degree; Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations.
Recommended reading: Sections 4.4, 5.1, 5.2
December 04: A graph \((V,E)\) is a tree iff it is connected and \(|E|=|V|-1\); 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 5.1, 6.1--6.3
December 11: 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
December 18: Graph score; Score theorem: \(D=(d_1,\ldots,d_n)\) is a graph score iff \(D'=(d_1,\ldots,d_{n-d_n-1},d_{n-d_n}-1,\ldots,d_{n-1}-1)\) is a graph score; Graph isomorphism on trees.
Recommended reading: Sections 4.3, 5.2
January 08:
End of 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; Basic notation \(\mathbb{N,Z,Q,R}\); Mathematical induction.
Recommended reading: Sections 1.1-1.3
October 09: Well-defined sets; writing a set; operations on sets: union, intersection, difference, complement; functions: injection, surjection, bijection; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive; Number of subsets of a set.
Recommended reading: Sections 1.4-1.6
October 16: Notation \(2^X\); number of subsets of a set via bijection with set binary strings; number of functions from an \(m\)-element set to an \(n\)-element set; number of injective functions functions from \([m]\) to \([n]\); 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).
Recommended reading: Sections 3.1-3.3
October 23: Inclusion-Exclusion principle (proof by double counting); Application: number of surjective functions. Orderings; linear/total vs. partial; Poset; minimal, maximal, minimum, maximum elements; Every finite poset has at least one minimal element.
Recommended reading: Sections 3.7, 2.1-2.2
October 30: Drawing posets: Hasse diagram; Chains, antichatins; Notation: \(\alpha(P),\omega(P)\); Theorem: \(\alpha(P)\omega(P)\geqslant |P|\); Application: Erdös-Szekeres theorem; Counting the number of nonnegative integer solutions of \(x_1+x_2+\ldots+x_r=n, l_i \leqslant x_i \leqslant u_i\); Equivalence relation: equivalence classes, partition of a set.
Recommended reading: Sections 2.3--2.4, 3.3, 1.6
November 06: Equivalence relation: partitition into equivalence classes; Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; Conditional probability.
Recommended reading: Sections 1.6, 10.2; Notes on Probability
November 13: 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; Indpendendent events; 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 bipartitite subgraph with at least half the number of edges (proof outline using expectation).
Recommended reading: Sections 10.2--10.4; Notes on Probability
November 20: Every graph has a bipartitite subgraph with at least half the number of edges (proof using expectation); Graph isomorphism; More examples of graphs: cycle, path, Handshake lemma; Subgraphs/Induced subgraphs; connectedness; walk; circuit; Eulerian graphs; A graph is Eulerian if and only if every vertex has even degree.
Recommended reading: Sections 4.1, 4.2, 4.4, 10.4
November 27: A graph is Eulerian if and only if every vertex has even degree; Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations.
Recommended reading: Sections 4.4, 5.1, 5.2
December 04: A graph \((V,E)\) is a tree iff it is connected and \(|E|=|V|-1\); 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 5.1, 6.1--6.3
December 11: 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
December 18: Graph score; Score theorem: \(D=(d_1,\ldots,d_n)\) is a graph score iff \(D'=(d_1,\ldots,d_{n-d_n-1},d_{n-d_n}-1,\ldots,d_{n-1}-1)\) is a graph score; Graph isomorphism on trees.
Recommended reading: Sections 4.3, 5.2
January 08:
End of lectures.