# Basic Principles

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```							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
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
Homework and preparation:   10%
Midterm Exam 1:             30%
Midterm Exam 2:             30%
Final Exam:                 30%

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
27

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