MATH-331 Discrete Mathematics Fall

Organizational Details Class Meeting: 11 :00am-12:15pm; Monday, Wednesday; Room SCIT215 Instructor: Dr. Igor Aizenberg Office: Science and Technology Building, 104C Phone ( ) Office hours: Monday, Wednesday 10am-6pm Tuesday 11pm-3pm Class Web Page : 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 ( ); Uzhgorod National University (Ukraine, and ); Catholic University of Leuven (Belgium, ); Company “Neural Networks Technologies” (Israel, ); University of Dortmund (Germany, ); National Center of Advanced Industrial Science and Technologies (Japan, 2004); Tampere University of Technology (Finland, ); Texas A&M University-Texarkana, from March,

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

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 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 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 10

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 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): For example, 12

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 ? 13

Subsets 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 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: 18

Operations on Sets: Union Important property: 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): When the sets A and B are said to be mutually exclusive or disjoint. 20

Operations on Sets: Intersection Important property: 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: 22

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. 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: Associative Laws: Distributive Laws: 25

Algebra of Sets Complementary: Difference Laws: 26

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