Discrete Mathematics

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Syllabus

Introduction - Basic Objects in Discrete Mathematics

This module gives the learner a first impression of what discrete mathematics is about, and in which ways its "flavor" differs from other fields of mathematics. It introduces basic objects like sets, relations, functions, which form the foundation of discrete mathematics.

Even without knowing, the learner has seen some orderings in the past. Numbers are ordered by <=. Integers can be partially ordered by the "divisible by" relation. In genealogy, people are ordered by the "A is an ancestor of B" relation. This module formally introduces partial orders and proves some fundamental and non-trivial facts about them.

A big part of discrete mathematics is about counting things. A classic example asks how many different words can be obtained by re-ordering the letters in the word Mississippi. Counting problems of this flavor abound in discrete mathematics discrete probability and also in the analysis of algorithms.

The binomial coefficient (n choose k) counts the number of ways to select k elements from a set of size n. It appears all the time in enumerative combinatorics. A good understanding of (n choose k) is also extremely helpful for analysis of algorithms.

Graphs are arguably the most important object in discrete mathematics. A huge number of problems from computer science and combinatorics can be modelled in the language of graphs. This module introduces the basic notions of graph theory - graphs, cycles, paths, degree, isomorphism.

We continue with graph theory basics. In this module, we introduce trees, an important class of graphs, and several equivalent characterizations of trees. Finally, we present an efficient algorithm for detecting whether two trees are isomorphic.

Starting with the well-known "Bridges of Königsberg" riddle, we prove the well-known characterization of Eulerian graphs. We discuss Hamiltonian paths and give sufficient criteria for their existence with Dirac's and Ore's theorem.

We discuss spanning trees of graphs. In particular we present Kruskal's algorithm for finding the minimum spanning tree of a graph with edge costs. We prove Cayley's formula, stating that the complete graph on n vertices has n^(n-2) spanning trees.

This module is about flow networks and has a distinctively algorithmic flavor. We prove the maximum flow minimum cut duality theorem.

We prove Hall's Theorem and Kőnig's Theorem, two important results on matchings in bipartite graphs. With the machinery from flow networks, both have quite direct proofs. Finally, partial orderings have their comeback with Dilworth's Theorem, which has a surprising proof using Kőnig's Theorem.