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Discrete Mathematics
  • Level Professional
  • Duration 42 hours
  • Course by Shanghai Jiao Tong University
  • Offered by

About

Discrete mathematics forms the mathematical foundation of computer and information science. It is also a fascinating subject in itself. Learners will become familiar with a broad range of mathematical objects like sets, functions, relations, graphs, that are omnipresent in computer science. Perhaps more importantly, they will reach a certain level of mathematical maturity - being able to understand formal statements and their proofs; coming up with rigorous proofs themselves; and coming up with interesting results. This course attempts to be rigorous without being overly formal. This means, for every concept we introduce we will show at least one interesting and non-trivial result and give a full proof. However, we will do so without too much formal notation, employing examples and figures whenever possible. The main topics of this course are (1) sets, functions, relations, (2) enumerative combinatorics, (3) graph theory, (4) network flow and matchings. It does not cover modular arithmetic, algebra, and logic, since these topics have a slightly different flavor and because there are already several courses on Coursera specifically on these topics.

Modules

Introduction to the course

    • 1 Videos
    • 1 PeerReview
Show info about module content

1 Peer Review

  • Exercises for introduction lesson

1 Videos

  • Introduction to the course

Sets, Relations, Functions

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Sets, relations, and functions

1 Peer Review

  • Sets, Relations, Functions

1 Videos

  • Sets, Relations, Functions

Partial orders, maximal and minimal elements, chains, antichains

    • 1 Videos
Show info about module content

1 Videos

  • Partial orderings: basic notions

Partial Orders: Mirsky's and Dilworth's Theorem

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Partial orders, maximal and minimal elements, chains, antichains

1 Peer Review

  • Partial orders, maximal and minimal elements, chains, antichains

1 Videos

  • Mirsky's and Dilworth's Theorem

How to Count Functions, Injections, Permutations, and Subsets

    • 1 Videos
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1 Videos

  • How to Count Functions, Injections, Permutations, and Subsets

Evaluating Simple Sums

    • 1 Videos
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1 Videos

  • Evaluating Simple Sums

Pascal's Triangle (or Yang Hui triangle or Al-Khayyam triangle) and a Recurrence for the Binomial Coefficient

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Counting Basic Objects

1 Peer Review

  • Counting Basic Objects

1 Videos

  • Pascal's Triangle

Combinatorial Identities

    • 1 Videos
    • 2 PeerReview
    • 1 Assignment
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1 Assignment

  • An Eagle's View of Pascal's Triangle

2 Peer Review

  • Combinatorial Identities
  • Digging Into Pascal's Triangle

1 Videos

  • Combinatorial Identities

Estimating the Binomial Coefficient

    • 1 Videos
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1 Videos

  • Estimating the Binomial Coefficient

Excursion to Discrete Probability: Computing the Expected Minimum of k Random Elements from {1,...,n}

    • 1 Videos
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1 Videos

  • Excursion to Discrete Probability: Computing the Expected Minimum of k Random Elements from {1,...,n}

Asymptotics and the O( )-Notation

    • 1 Videos
    • 2 PeerReview
    • 1 Assignment
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1 Assignment

  • The Big-O-Notation

2 Peer Review

  • Basic Facts
  • Classes that often occur in complexity theory

1 Videos

  • Asymptotics and the O( )-Notation

Basic Notions and Examples

    • 1 Videos
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1 Videos

  • Basic Notions and Examples

Graph Isomorphism, Degree, Graph Score

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Graphs, isomorphisms, and the sliding tile puzzle

1 Peer Review

  • Graphs and Isomorphisms

1 Videos

  • Graph Isomorphism, Degree, Graph Score

Graph Score Theorem

    • 1 Videos
    • 1 PeerReview
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1 Peer Review

  • The Graph Score Theorem

1 Videos

  • Graph Score Theorem

Graph Connectivity

    • 1 Videos
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1 Videos

  • Graphs and Connectivity

Cycles and Trees

    • 1 Videos
    • 2 PeerReview
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2 Peer Review

  • Cycles and Trees
  • Spanning Tree Exchange Graph

1 Videos

  • Cycles and Trees

An Efficient Algorithm for Isomorphism of Trees

    • 1 Videos
    • 1 Assignment
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1 Assignment

  • Cycles and Trees

1 Videos

  • An Efficient Algorithm for Isomorphism of Trees

Eulerian Cycles

    • 1 Videos
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1 Videos

  • Eulerian Cycles

Hamilton Cycles

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Hamiltonian Cycles and Paths

1 Peer Review

  • Hamiltonian Cycles and Paths

1 Videos

  • Hamilton Cycles - Ore's and Dirac's Theorem

Minimum Spanning Trees

    • 1 Videos
    • 1 PeerReview
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1 Peer Review

  • Minimum Spanning Trees

1 Videos

  • Minimum Spanning Trees

The Number of Trees on n Vertices

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Spanning Trees

1 Peer Review

  • Counting Trees on n Vertices

1 Videos

  • The Number of Trees on n Vertices

Flow Networks, Flows, Cuts: Basic Notions and Examples

    • 1 Videos
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1 Videos

  • Flow Networks, Flows, Cuts: Basic Notions and Examples

Flow Networks: The Maxflow - Mincut Theorem

    • 1 Videos
    • 1 PeerReview
    • 1 Assignment
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1 Assignment

  • Network flow

1 Peer Review

  • Network Flows

1 Videos

  • Flow Networks: The Maxflow - Mincut Theorem

Matchings in Bipartite Graphs: Basic Notions and an Algoriths

    • 1 Videos
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1 Videos

  • Matchings in Bipartite Graphs - Basic Notions and an Algorithm

Matchings in Bipartite Graphs: Hall's and König's Theorem

    • 1 Videos
    • 1 PeerReview
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1 Peer Review

  • Matchings in Bipartite Graphs

1 Videos

  • Matchings in Bipartite Graphs: Hall's and König's Theorem

Partial Orders: Dilworth's Theorem on Chains and Antichains

    • 1 Videos
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1 Videos

  • Partial Orders: Dilworth's Theorem on Chains and Antichains

Auto Summary

"Discrete Mathematics" is an essential course for those interested in the mathematical foundations of computer and information science. Taught by Coursera, the course covers sets, functions, relations, combinatorics, graph theory, and network flow. It emphasizes understanding formal statements, proofs, and mathematical maturity through rigorous yet accessible content. Suitable for professional learners, it offers a comprehensive 2520-minute curriculum with Starter and Professional subscription options. Ideal for aspiring computer scientists and mathematicians seeking to deepen their knowledge.

Dominik Scheder
Instructor

Dominik Scheder