Basic Principles

							MATH-331 Discrete Mathematics

           Fall 2009




                                1
          Organizational Details
Class Meeting:
11:00am-12:15pm; Monday, Wednesday; Room SCIT215

Instructor: Dr. Igor Aizenberg

Office: Science and Technology Building, 104C
Phone (903 334 6654)
e-mail: igor.aizenberg@tamut.edu

Office hours:
Monday, Wednesday 10am-6pm
Tuesday 11pm-3pm

Class Web Page: http://www.eagle.tamut.edu/faculty/igor/MATH-331.htm


                                                                   2
 Dr. Igor Aizenberg: self-introduction

• MS in Mathematics from Uzhgorod National University (Ukraine),
  1982
• PhD in Computer Science from the Russian Academy of Sciences,
  Moscow (Russia), 1986
• Areas of research: Artificial Neural Networks, Image Processing and
  Pattern Recognition
• About 100 journal and conference proceedings publications and
  one monograph book
• Job experience: Russian Academy of Sciences (1982-1990);
  Uzhgorod National University (Ukraine,1990-1996 and 1998-1999);
  Catholic University of Leuven (Belgium, 1996-1998); Company
  “Neural Networks Technologies” (Israel, 1999-2002); University of
  Dortmund (Germany, 2003-2005); National Center of Advanced
  Industrial Science and Technologies (Japan, 2004); Tampere
  University of Technology (Finland, 2005-2006); Texas A&M
  University-Texarkana, from March, 2006
                                                                        3
                 Text Book

• "Discrete Mathematics" by J. A. Dossey, A.
  D. Otto, L. E. Spence, and C. V. Eynden, 5th
  Edn., Pearson/Addison Wesley, 2006, ISBN
  0-321-30515-9.




                                                 4
          Control

Exams (open book, open notes):
Exam 1:    October 5-7, 2009
Exam 2:    November 4, 2009
Exam 3:    December 14, 2009


Homework



                                  5
             Grading
Grading Method
Homework and preparation:   10%
Midterm Exam 1:             30%
Midterm Exam 2:             30%
Final Exam:                 30%

Grading Scale:
90%+  A
80%+  B
70%+  C
60%+  D
less than 60%  F
                                  6
           What we will study?

• Basic concepts of discrete mathematics,
  which are used in computer modeling and
  simulation
• Combinatorics
• Probability theory
• Function theory
• Graph theory
• Encoding theory
• Elements of Mathematical Logic
                                            7
                      Sets
• The set, in mathematics, is any collection of
  objects of any nature specified according to a
  well-defined rule.
• Each object in a set is called an element (a
  member, a point). If x is an element of the set
  X, (x belongs to X) this is expressed by
                     x X
•   x X    means that x does not belong to X

                                                    8
                       Sets
• Sets can be finite (the set of students in the
  class), infinite (the set of real numbers) or empty
  (null - a set of no elements).
• A set can be specified by either giving all its
  elements in braces (a small finite set) or stating
  the requirements for the elements belonging to
  the set.
• X={a, b, c, d}
• X={x : x is a student taking the “Discrete
  Mathematics” class}

                                                        9
                       Sets
•   X Z    is the set of integer numbers
•   X Q    is the set of rational numbers
•   X R    is the set of real numbers
•   X C    is the set of complex numbers
•      is an empty set
•   X   is a set whose single element is an
           empty set


                                                  10
                     Sets
• What about a set of the roots of the equation

                 2 x 2  1  0?

• The set of the real roots is empty: 
                                             
• The set of the complex roots is i / 2, i / 2 ,
  where i is an imaginary unity



                                                  11
                  Subsets
• When every element of a set A is at the same
  time an element of a set B then A is a subset
  of B (A is contained in B):
                     A B
                     B A
• For example,
          Z  Q, Z  R, Q  R, R  C

                                                  12
                    Subsets
• The sets A and B are said to be equal if they
  consist of exactly the same elements.
• That is, A  B, B  A  A  B
• For instance, let the set A consists of the roots
  of equation
                 x( x  1)( x  4)( x  3)  0
                           2


                 B  2, 1, 0, 2,3
                 C   x | x  Z,| x | 4
• What about the relationships among A, B, C ?
                                                  13
          Subsets
AC
BC
A  B
      A B
B  A
A :   A but in general   A


                                  14
                 Cardinality
• The cardinality of a finite set is the number of
  elements in the set.
• |A| is the cardinality of A.
• A set with the same cardinality as any subset
  of the set of natural numbers is called a
  countable set.



                                                     15
                 Continuum
• For an infinite continuous linearly ordered set,
  which has a property that it there is always an
  element between other two, we say that its
  cardinality is “continuum” (for example,
  interval [0,1], any other interval, or any line)
  or simply call this set continuum.




                                                 16
               Universal Set
• A large set, which includes some useful in
  dealing with the specific problem smaller sets,
  is called the universal set (universe). It is
  usually denoted by U.
• For instance, in the previous example, the set
  of integer numbers Z can be naturally
  considered as the universal set.


                                                17
      Operations on Sets: Union
• Let U be a universal set of any arbitrary
  elements and contains all possible elements
  under consideration. The universal set may
  contain a number of subsets A, B, C, D, which
  individually are well-defined.
• The union (sum) of two sets A and B is the set
  of all those elements that belong to A or B or
  both:
                     A      B
                                               18
   Operations on Sets: Union
 A  {a, b, c, d }; B  {e, f }; A B  {a, b, c, d , e, f }
 A  {a, b, c, d }; B  {c, d , e, f }; A B  {a, b, c, d , e, f }
 A  {a, b, c, d }; B  {c, d }; A B  {a, b, c, d}  A


Important property:

           B  A A B  A

                                                                19
  Operations on Sets: Intersection
• The intersection (product) of two sets A and B
  is the set of all those elements that belong to
  both A and B (that are common for these
  sets):
                    A      B

• When A B   the sets A and B are said to
  be mutually exclusive or disjoint.

                                                20
Operations on Sets: Intersection
     A  {a, b, c, d }; B  {e, f }; A B  
     A  {a, b, c, d }; B  {c, d , f }; A B  {c, d }
     A  {a, b, c, d }; B  {c, d }; A B  {c, d }  B


Important property:

        B  A A B  B

                                                         21
   Operations on Sets: Difference
• The difference of two sets A and B is the set of
  all those elements that belong to the set A but
  do not belong to the set B:

    A / B or A  B
    A  B   x : x  A and x  B


                                                 22
 Operations on Sets: Complement
• The complement (negation) of any set A is the
  set A’ ( A ) containing all elements of the
  universe that are not elements of A.




                                              23
              Venn Diagrams
• A Venn diagram is a useful mean for
  representing relationships among sets (see p.
  44 in the text book) .
• In a Venn diagram, the universal set is
  represented by a rectangular region, and
  subsets of the universal set are represented by
  circular discs drawn within the rectangular
  region.

                                                24
              Algebra of Sets
• Let A, B, and C be subsets of a universal set U.
  Then the following laws hold.
• Commutative Laws: A  B  B  A; A  B  B  A
• Associative Laws: ( A  B)  C  A  ( B  C )
                       ( A  B)  C  A  ( B  C )
• Distributive Laws:
                       A  ( B  C )  ( A  B)  ( A  C )
                       A  ( B  C )  ( A  B)  ( A  C )


                                                              25
             Algebra of Sets
                     A A U
• Complementary:     A A  
                     A U  U
                     A U  A
                     A  A
                     A  

• Difference Laws:
                   ( A  B)  ( A  B)  A
                   ( A  B)  ( A  B)  
                   A B  A B
                                             26
             Algebra of Sets
• De Morgan’s Laws (Dualization):
                A  B  A  B
                A  B  A  B
                  
• Involution Law: A  A
• Idempotent Law: For any set A:
                   A A  A
                   A A  A
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