MATH-331 Discrete Mathematics

Fall 2009

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

Instructor: Dr. Igor Aizenberg

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

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

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

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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

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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.

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Control

Exams (open book, open notes):

Exam 1: October 5-7, 2009

Exam 2: November 4, 2009

Exam 3: December 14, 2009

Homework

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Grading

Grading MethodHomework 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

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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

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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

• means that x does not belong to Xx X

x X

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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}

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Sets• is the set of integer numbers• is the set of rational numbers• is the set of real numbers• is the set of complex numbers• is an empty set• is a set whose single element is an empty

set

X ZX QX RX C

X

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Sets

• What about a set of the roots of the equation

• The set of the real roots is empty:• The set of the complex roots is ,

where i is an imaginary unity

22 1 0?x

/ 2, / 2i i

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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):

• For example,

A BB A

, , , Z Q Z R Q R R C

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Subsets• The sets A and B are said to be equal if they consist

of exactly the same elements.• That is,• For instance, let the set A consists of the roots of

equation

• What about the relationships among A, B, C ?

,A B B A A B

2( 1)( 4)( 3) 0

2, 1,0,2,3

| ,| | 4

x x x x

B

C x x x

Z

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Subsets

: but in general

A C

B C

A BA B

B A

A A A

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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.

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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.

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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 can be naturally considered as the universal set.

Z

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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

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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

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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):

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

A B

A B

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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

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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:

or

: and

/A B A B

A B x x A x B

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Operations on Sets: Complement

• The complement (negation) of any set A is the set A’ ( ) containing all elements of the universe that are not elements of A.

A

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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.

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Algebra of Sets

• Let A, B, and C be subsets of a universal set U. Then the following laws hold.

• Commutative Laws:• Associative Laws:

• Distributive Laws:

;A B B A A B B A

( ) ( )

( ) ( )

A B C A B C

A B C A B C

( ) ( ) ( )

( ) ( ) ( )

A B C A B A C

A B C A B A C

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Algebra of Sets

• Complementary:

• Difference Laws:

A A U

A A

A U U

A U A

A A

A

( ) ( )

( ) ( )

A B A B A

A B A B

A B A B

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Algebra of Sets• De Morgan’s Laws (Dualization):

• Involution Law: • Idempotent Law: For any set A:

A B A B

A B A B

A A

A A A

A A A

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