Difference between revisions of "MAT5433"

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Introduction to basic discrete structures.
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== Catalog entry ==
  
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'''Sample textbooks''':  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
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''Contents''
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(1) Basic counting principles: Permutations, combinations, binomial coefficients, arrangements with repetitions.
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(2) The Inclusion-Exclusion principle.
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(3) Graph models: Isomorphisms, edge counting, planar graphs.
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(4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem
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(5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems.
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(6) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles.
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== Sample textbooks ==
  
 
[1] Vladlen Koltun, ''Discrete Structures, Lecture Notes, Stanford University'', 2008. Freely available online [https://web.stanford.edu/class/cs103x/cs103x-notes.pdf here]
 
[1] Vladlen Koltun, ''Discrete Structures, Lecture Notes, Stanford University'', 2008. Freely available online [https://web.stanford.edu/class/cs103x/cs103x-notes.pdf here]
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'''Catalog entry'''
 
  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
 
  
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''Contents''
 
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(1) Relations: Cartesian products, relations, properties of relations, equivalence relations and partitions. (2) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles, Zorn's Lemma, lattices, boolean algebras, circuit design. (3) Graphs: Euler and Hamiltonian paths and circuits, matching, graph coloring, Ramsey’s theorem, trees and searching. (4) Binary operations: Groups and semigroups, products and quotients of groups, other algebraic structures.
 
  
  
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|  1-2   
 
|  1-2   
 
|| [[Basic counting principles]]
 
|| [[Basic counting principles]]
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|| 10.1-10-2
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|| 5.1-5.5
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|| Permutations, combinations, arrangements with repetitions
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|| Permutations, combinations, binomial coefficients, arrangements with repetitions
 
|| MAT1313, CS2233/2231, or instructor consent.
 
|| MAT1313, CS2233/2231, or instructor consent.
 
|-
 
|-
 
|  3   
 
|  3   
 
|| [[Inclusion-Exclusion Principle]]
 
|| [[Inclusion-Exclusion Principle]]
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|| 2.5-2.7
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|| 8.1-8.2
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|| Counting with Venn diagrams.
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||  
 
 
|-
 
|-
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|  4   
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|  4-6    
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|| [[The Pigeonhole Principle]]
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|| [[Graph models]]
 
|| 12.1-12.3
 
|| 12.1-12.3
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||  
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|| Isomorphism, edge counting, planar graphs.
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|-
 
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|  Graphs 
 
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|| [[Asymptotics]]
 
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|| 13.1-13.2
 
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||
 
 
|-
 
|-
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|  8   
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|  7-8   
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|| [[Relations]]  
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|| [[Covering circuits]]
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|| 5.1-6.3
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|| 2.1-2.4
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||  
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|| Euler circuits, Hamilton circuits, graph colorings, coloring theorems.
 
|-
 
|-
 
|  9-10   
 
|  9-10   
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|| [[Discrete structures (graphs, partially ordered sets) ]]  
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|| [[Trees]]  
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|| 7.1-8.4
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|| 3.1-3.4
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||  
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|| Search trees, spanning trees, the Traveling Salesman Problem
 
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|-
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|  10-16    
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|  11-13    
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|| [[Models of computation]]  
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|| [[Network algorithms]]  
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|| 10.1-10.4
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|| 4.1-4.5
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||  
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|| Shortest path , minimum spanning trees, matching algorithms, transportation problems.
 
|}
 
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Latest revision as of 22:09, 25 March 2023

Catalog entry

Prerequisite: Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

Contents (1) Basic counting principles: Permutations, combinations, binomial coefficients, arrangements with repetitions. (2) The Inclusion-Exclusion principle. (3) Graph models: Isomorphisms, edge counting, planar graphs. (4) Covering circuits and graph colorings: Euler circuits, Hamilton circuits, graph colorings, Ramsey's theorem (5) Network algorithms: Shortest path, minimum spanning trees, matching algorithms, transportation problems. (6) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles.

Sample textbooks

[1] Vladlen Koltun, Discrete Structures, Lecture Notes, Stanford University, 2008. Freely available online here

[2] Alan Tucker, Applied Combinatorics, Freely available online here






Topics List

Week Topic Sections from Tucker's book Subtopics Prerequisite
1-2 Basic counting principles 5.1-5.5 Permutations, combinations, binomial coefficients, arrangements with repetitions MAT1313, CS2233/2231, or instructor consent.
3 Inclusion-Exclusion Principle 8.1-8.2 Counting with Venn diagrams.
4-6 Graph models 12.1-12.3 Isomorphism, edge counting, planar graphs.
7-8 Covering circuits 2.1-2.4 Euler circuits, Hamilton circuits, graph colorings, coloring theorems.
9-10 Trees 3.1-3.4 Search trees, spanning trees, the Traveling Salesman Problem
11-13 Network algorithms 4.1-4.5 Shortest path , minimum spanning trees, matching algorithms, transportation problems.
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