# Discrete Mathematics

This is a basic course for undergraduate students.### Organization

For the Winter Semester 2019-2020, the lectures are scheduled on Tuesdays at 10:40 am in lecture room T8 at Troja. Tutorials are held immediately afterwards.## 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

Follow this link to see what was covered during the past years in this course. The exact material covered during the lectures will be updated on this page. The 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 01:**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.

**October 08:**Mathematical induction (weak/strong); Well-defined sets; Basic notation \(\mathbb{N,Z,Q,R}\); writing a set; operations on sets: union, intersection, difference, complement; Power set: Notation \(2^X\); Number of subsets of a set: proof by induction; ordered pair, tuples; cartesian product, \(n\)-fold product; relations: reflexive, symmetric, anti-symmetric, transitive.

**October 15:**Inverse of a relation; composition of relations; functions: injection, surjection, bijection; number of functions from an \(m\)-element set to an \(n\)-element set; number of injective functions functions from \([m]\) to \([n]\); pigeonhole principle; permutations: two-line and one-line notation; number of permutations of a finite set.

**October 22:**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 \(x_1+x_2+\ldots+x_r=n\); binomial theorem (proof by induction); applications: \(\sum_{k=0}^n {n\choose k}=2^n\), number of even and odd cardinality subsets.

**November 05:**Equivalence relation; partitition into equivalence classes; example: \(x\sim y\) iff \( x=y\text{ mod }5\); Inclusion-Exclusion principle (proof by double counting); Application: number of surjective functions.

**November 19:**Order; linear/total vs. partial; Poset; minimal, maximal, minimum, maximum elements; Every finite poset has at least one minimal element; Drawing posets: Hasse diagram; Chains, antichatins; Notation: \(\alpha(P),\omega(P)\); Theorem: \(\alpha(P)\omega(P)\geqslant |P|\).

**November 26:**Definitions: Probability space, uniform probability; Example: Probability of getting exactly \(k\) heads in the throw of \(n\) coins; 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; Conditional probability; Baye's theorem; Independent events.

**December 03:**Definitions: Random variable, expectation; Indicator function; Linearity of expectation, independent random variables; 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 using expectation).

**December 10:**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.

**December 17 (MK):**Connectedness; Connected graph and components; Trees; End-vertex lemma; Tree-growing lemma; five different tree characterizations.

**January 07:**Topological graph; 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; Kuratowski theorem (without proof); Proper coloring of a graph; Chromatic number.

**End of lectures.**