Shanghai Jiao Tong University

Discrete Mathematics

Taught in English

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51,574 already enrolled

Course

Gain insight into a topic and learn the fundamentals

Dominik Scheder

Instructor: Dominik Scheder

3.2

(175 reviews)

Intermediate level
Some related experience required
41 hours to complete
3 weeks at 13 hours a week
Flexible schedule
Learn at your own pace

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10 quizzes

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There are 11 modules in this course

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.

What's included

2 videos1 quiz2 peer reviews

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.

What's included

2 videos1 quiz1 peer review

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.

What's included

3 videos1 quiz1 peer review

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.

What's included

3 videos1 quiz2 peer reviews

What's included

1 video1 quiz2 peer reviews

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.

What's included

3 videos1 quiz2 peer reviews

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.

What's included

3 videos1 quiz2 peer reviews

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.

What's included

2 videos1 quiz1 peer review

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.

What's included

2 videos1 quiz2 peer reviews

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

What's included

2 videos1 quiz1 peer review

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.

What's included

3 videos1 peer review

Instructor

Instructor ratings
3.0 (14 ratings)
Dominik Scheder
Shanghai Jiao Tong University
1 Course51,574 learners

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3.2

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