About this Course
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Intermediate Level

Approx. 40 hours to complete

Suggested: 11 weeks of study, 3-5 hours per week....

English

Subtitles: English

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Intermediate Level

Approx. 40 hours to complete

Suggested: 11 weeks of study, 3-5 hours per week....

English

Subtitles: English

Syllabus - What you will learn from this course

Week
1
5 hours to complete

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....
2 videos (Total 27 min), 3 quizzes
2 videos
Sets, Relations, Functions10m
1 practice exercise
Sets, relations, and functions30m
Week
2
4 hours to complete

Partial Orders

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....
2 videos (Total 28 min), 2 quizzes
2 videos
Mirsky's and Dilworth's Theorem14m
1 practice exercise
Partial orders, maximal and minimal elements, chains, antichainss
Week
3
5 hours to complete

Enumerative Combinatorics

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....
3 videos (Total 35 min), 2 quizzes
3 videos
Evaluating Simple Sums8m
Pascal's Triangle14m
1 practice exercise
Counting Basic Objectss
Week
4
4 hours to complete

The Binomial Coefficient

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....
3 videos (Total 55 min), 3 quizzes
3 videos
Estimating the Binomial Coefficient22m
Excursion to Discrete Probability: Computing the Expected Minimum of k Random Elements from {1,...,n}18m
1 practice exercise
An Eagle's View of Pascal's Triangle8m
Week
5
5 hours to complete

Asymptotics and the O-Notation

...
1 video (Total 14 min), 3 quizzes
1 practice exercise
The Big-O-Notation18m
Week
6
5 hours to complete

Introduction to Graph Theory

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....
3 videos (Total 41 min), 3 quizzes
3 videos
Graph Isomorphism, Degree, Graph Score13m
Graph Score Theorem16m
1 practice exercise
Graphs, isomorphisms, and the sliding tile puzzle30m
Week
7
5 hours to complete

Connectivity, Trees, Cycles

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....
3 videos (Total 36 min), 3 quizzes
3 videos
Cycles and Trees15m
An Efficient Algorithm for Isomorphism of Trees12m
1 practice exercise
Cycles and Trees30m
Week
8
3 hours to complete

Eulerian and Hamiltonian Cycles

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....
2 videos (Total 27 min), 2 quizzes
2 videos
Hamilton Cycles - Ore's and Dirac's Theorem16m
1 practice exercise
Hamiltonian Cycles and Pathss
Week
9
5 hours to complete

Spanning Trees

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....
2 videos (Total 29 min), 3 quizzes
2 videos
The Number of Trees on n Vertices15m
1 practice exercise
Spanning Trees40m
Week
10
3 hours to complete

Maximum flow and minimum cut

This module is about flow networks and has a distinctively algorithmic flavor. We prove the maximum flow minimum cut duality theorem....
2 videos (Total 29 min), 2 quizzes
2 videos
Flow Networks: The Maxflow - Mincut Theorem15m
1 practice exercise
Network flow8m
Week
11
3 hours to complete

Matchings in Bipartite Graphs

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....
3 videos (Total 46 min), 1 quiz
3 videos
Matchings in Bipartite Graphs: Hall's and König's Theorem16m
Partial Orders: Dilworth's Theorem on Chains and Antichains15m
3.6
31 ReviewsChevron Right

Top Reviews

By NPOct 23rd 2017

Fantastic course. Fascinating material, presented at a reasonably fast pace, and some really challenging assignments.

By AGDec 5th 2018

This course is good to comprehend relation, function and combinations.

Instructor

Avatar

Dominik Scheder

Assistant Professor
The Department of Computer Science and Engineering

About Shanghai Jiao Tong University

Shanghai Jiao Tong University, a leading research university located in Shanghai, China, has been regarded as the fastest developing university in the country for the last decade. With special strengths in engineering, science, medicine and business, it now offers a comprehensive range of disciplines in 27 schools with more than 41,000 enrolled students from more than one hundred countries....

Frequently Asked Questions

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