MATH-331 Discrete Mathematics

1. MATH-331 Discrete Mathematics Fall 2009

2. 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

3. 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

4. "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. Text Book

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

6. 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

7. 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

8. 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 X

9. 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}

10. 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

11. 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

12. 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,

13. 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 ?

14. Subsets

15. 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.

16. 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.

17. 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.

18. 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:

19. Operations on Sets: Union Important property:

20. 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.

21. Operations on Sets: Intersection Important property:

22. 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:

23. 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.

24. 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.

25. 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:

26. Algebra of Sets • Complementary: • Difference Laws:

27. Algebra of Sets • De Morgan’s Laws (Dualization): • Involution Law: • Idempotent Law: For any set A: