BRONX COMMUNITY COLLEGE

Of the City University of New York

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

SYLLABUS: CSI30 DISCRETE MATHEMATICS 1 3 credits / 3 hours

PREREQUISITE: MTH 06 and ENG 02 and RDL 02 if required

TEXT: Discrete Mathematics and its Applications, Sixth Edition,

by Kenneth H. Rosen, published by McGraw-Hill 2007

Goals of the course:

CSI 30 is an introduction to mathematical methods in computer science. It begins with basic concepts of mathematical logic, continues with an introduction to algorithms and programming, and concludes with an introduction to counting techniques and probability. The emphasis is on computational, hands-on experience. The material on set theory reinforces and complements parallel topics covered in Precalculus (MTH 30). It is highly recommended that MTH 30, if required, and CSI 30 are taken in the same semester.

Objectives: A successful student in this course will learn to:

1. Understand the idea of an algorithm and computer program;

2. Write and analyze simple programs;

3. Understand the use of formal logic in mathematics and programming;

4. Understand basic concepts of set theory, particularly those of a function;

5. Use basic combinatorial counting techniques, particularly permutations and combinations;

6. Understand basic concepts of probability theory, and the way counting techniques are used there.

 Chapters and sections Suggested in-class examples Suggested Homework

Chapter 1 The Foundations: Logic and Proof, Sets, and Functions (5 weeks)

 1.1 Propositions. Boolean connectives. Implications. Examples 1-11 16/1, 3, 5, 7, 17 1.1 Translating English sentences.System specifications. BooleanSearches. Bit operations. TruthTables. Examples 12-21 19/27, 29, 37, 49, 53 1.2 Propositional Equivalences Examples All 28/1-9 (odd) 1.3 Predicates and Quantifiers Examples 1-12 Examples 23, 24, 28 46/l-25 (odd) 48/27, 31, 35, 55 1.4 Nested quantifiers. Examples 1-15 58/1, 3, 5, 9, 15, 25, 27, 33 1.5 Rules of Inference. Fallacies. Examples 1-11 72/1-7 (odd)

Chapter 2 Basic Structures: Sets, Functions, Sequences, Sums

 2.1 Sets, power sets, Cartesian products. Examples 1-19 119/l-9 (odd), 19-25 (odd) 2.2 Set operations. Set identities. Examples 1-14 130/1, 3, 25 2.2 Computer representations of sets. Examples 18, 19, 20 132/50, 51, 52, 53 2.3 One-to-one and onto functions. Examples 1-15 148/1-7 (odd) 2.3 Inverse and composition of functions. Graphs. Some important functions. Examples 16-28 148/9, 12, 13, 15, 19, 27, 32, 39

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 Chapters and sections Suggested in-class examples Suggested Homework

Chapter 3 The Fundamentals: Algorithms, the Integers, and Matrices

 3.1 Algorithms. Pseudocode. Searching algorithms Examples 1, 2 177/1, 3, 5 3.1 Sorting. Greedy algorithms. Examples 4, 5 177/2, 7, 13, 19, 35 3.4 Division. Examples 1-4 208/1, 9 3.4 Modular arithmetic. Applications of congruences. Pseudorandom numbers. Examples 5-8 208/17, 19, 27 3.5 Fundamental Theorem of Arithmetic. The Infinitude of Primes. Examples 1-5 217/3, 11, 12 3.5 Representations of integers Examples 1-6 229/1, 3, 5 3.5 Algorithms for integer operations. Modular exponentiation. The Euclidean algorithm. Examples 7-9, 11, 12 230/19, 23 3.8 Matrix arithmetic. Examples 1-6 254/1, 3, 5 3.8 Transposes and powers of Matrices. Zero-one matrices. Examples 7-12 255/19, 20, 28, 29

Chapter 5 Counting (4 weeks)

 5.1 Basic counting principles Examples 1-13 344/l-17 (odd) 5.1 More complex counting problems. Exclusion inclusion principle. Tree diagrams. Examples 14-21 345/19-31 (odd) 5.3 Permutations and combinations. Examples 1-15 360/1-19 (odd) 5.4 Binomial coefficients. Pascal's triangle. Examples 1-4 369/1-9 (odd), 12, 13

Chapter 6 Discrete Probability

 6.1 Introduction to probability Examples 1-9 398/l-27 (odd)

RK/2003; Revised Nov 2006/JP/SP Fall 2007, CO'S Fall 2008