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									                        Discrete Mathematics




Discrete Mathematics

   Cheng-Chia Chen

    September 2009

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                                             Discrete mathematics
  Course information:

l Textbook:
  Discrete Mathematics and its applications, by
  Kenneth H. Rosen (歐亞書局: TEL: 8912-1188)

Time: 13:10~16:00 Tuesday
l Room:大仁 3301
l Grading:
  ￿Two examinations (60%)
  ￿assignments and quizzes (30%)
  ￿Classroom participation (10%)



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                                                Discrete mathematics
  Course outlines
l The foundations:
  ￿Logic, sets and functions
l The fundamental:
  ￿algorithms, the integers and matrices.
l Mathematical reasoning
  ￿Methods of proofs, Mathematical induction, recursive
  definitions, recursive algorithms, program correctness.
l Counting:
  ￿Basic counting method, pigeon hole principle,
   permutation and combinations.
l Advanced counting techniques:
  ￿Recurrence relations, generating functions, inclusion-
   exclusion principle, applications.

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                                                  Discrete mathematics
  course outline(cont’d)
l Relations:
  ￿definition, property of relations, closure of relations,
  ￿kinds of relations(equivalence, partial order, lattices,..)
l Graphs:
  ￿ terminology, representation, connectivity,
  ￿ Euler and Hamilton paths, shortest path problems,
  ￿ planar graphs, graph coloring
l Trees:
  ￿terminology, applications
  ￿tree traversal, trees and sorting,
  ￿spanning trees, minimum spanning trees



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                                              Discrete mathematics
courses outline (cont’d)

l Modeling of computations
  ￿ Languages and grammars,
  ￿ FSM with O/P (Mealey and Moore machines),
  ￿ FSM w/o O/P,
  ￿CFL & CFG,
  ￿Turing machine.
l [Boolean algebra:]
l [Abstract algebra:
  ￿definition, monoid, group, ring, … ]




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                                                 Discrete mathematics
  Goal of this course
l Mathematical reasoning:
  ￿In order to read, comprehend and construct mathematical
   arguments.
l Combinatorial Analysis
l Discrete structures:
  ￿Familiar with math. structures used to represent discrete
   objects and relationships between these objects.
  ￿include sets, multisets, permutations, relations, graphs,
   trees and finite state machines.
  ￿groups, monoid(lists, or sequences), ring, field, lattices
   etc.
l Applications and modeling
l Algorithmic thinking: (constructive, procedural).
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