ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE. School of Mathematical Sciences

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! ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE School of Mathematical Sciences New Revised COURSE: COS-MATH-190H Honors Discrete Mathematics 1.

03-20-13 Optional Course Designations: Yes No General Education Writing Intensive Honors Approval Request Date Approval Grant Date 2.

1 ! ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE School of Mathematical Sciences New Revised COURSE: COS-MATH-190H Honors Discrete Mathematics 1.0 Course designations and approvals: Required Course Approvals: Approval Approval Request Date Grant Date Academic Unit Curriculum Committee College Curriculum Committee Optional Course Designations: Yes No General Education Writing Intensive Honors Approval Request Date Approval Grant Date 2.0 Course information: Course Title: Honors Discrete Mathematics Credit Hours: 3 Prerequisite(s): Honors status Co-requisite(s): None Course proposed by: School of Mathematical Sciences Effective date: Fall 2013 Contact Hours Maximum Students/section Classroom 3 35 Lab Workshop Other (specify) 2.1 Course conversion designation: (Please check which applies to this course) Semester Equivalent (SE) to: Semester Replacement (SR) to: and New 2.2 Semester(s) offered: Fall Spring Summer Offered every other year only Other Page 1 of 6

2 2.3 Student requirements: Students required to take this course: (by program and year, as appropriate) None Students who might elect to take the course: Honors students who wish to fulfill general education requirements with an honors mathematics course. Also students majoring in Applied Mathematics, Computational Mathematics, Bioinformatics, Computer Engineering, Computer Science, Software Engineering, and Applied Statistics, and seeking to strengthen their technical background in mathematics such as those pursuing a minor in mathematics. 3.0 Goals of the course: (including rationale for the course, when appropriate) 3.1 To introduce structures and fundamental techniques in discrete mathematics that are central to mathematics, computer science, and statistics. 3.2 To foster the skill of understanding and creating mathematically valid arguments. 3.3 To learn various methods of mathematical proof with emphasis on applications, and illustrations of these methods. 3.4 To learn how to read mathematics and how to write mathematics. 3.5 To develop skills in concise exposition, cogent communication of mathematical ideas and how to use them in computer applications. 3.6 To provide a background in mathematics which can be used for the study of science and engineering. 4.0 Course description: (as it will appear in the RIT Catalog, including pre- and co-requisites, semesters offered) COS-MATH-190H Honors Discrete Mathematics This course introduces students to widely used ideas and techniques from discrete mathematics. Students will learn about the fundamentals of propositional and predicate calculus, set theory, relations, recursive structures, counting techniques and their applications in advanced mathematics. This course is designed to challenge honors students by providing demanding problems and proofs in set theory, number theory, combinatorics and graph theory. Students will not only be exposed to these topics but will learn to think abstractly about them. Credit cannot be earned for this class if credit is earned in COS-MATH-190. (Pre-requisite: honors status.) Class 3, Credit 3 (F) 5.0 Possible resources: (texts, references, computer packages, etc.) 5.1 E. D. Bloch, Proofs and Fundamentals: A First Course in Abstract Mathematics, Birkhäuser Boston. 5.2 J. P. D Angelo and D. B. West, Mathematical Thinking: Problem-Solving and Proofs, Addison-Wesley. 5.3 G. Chartrand, A. D. Polimeni and P. Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, Addison Wesley. 5.4 R. Grimaldi, Discrete and Combinatorial Mathematics:An Applied Introduction, Addison- Wesley. Page 2 of 6

3 5.5 E. A. Scheinerman, Mathematics: A Discrete Introduction, Brooks Cole. 5.6 L. Lovász. J. Pelikán and K. Vesztergombi, Discrete Mathematics, Springer. 5.7 R. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison Wesley. 5.8 J. Hein, Discrete Structures, Logic, and Computability, Jones & Bartlett. 6.0 Topics: (outline) Topics with an asterisk(*) are at the instructor s discretion, as time permits 6.1 Logic Relations between statements Valid arguments Quantifiers 6.2 Strategies for Proofs Direct proofs Proofs by contrapositive and contradiction Cases and equivalent statements Quantifiers in Theorems Existence and uniqueness proofs Proving statements based on given definitions 6.3 Basic Set Theory Concept of: element of, subset of, and equality of sets Formal Representation of Set: Russell s Paradox and The Comprehension Axiom Operations on sets: Union, Intersection, Complement, Symmetric Difference Indexed families of sets Set Equalities: Logically Equivalent Products and The Element Method Notion of Power Set and Algorithm for its Enumeration Partitions of a Set Cartesian Product 6.4 Relations Relations as Cartesian product Ordering relations Partially and Totally Ordered Sets Topological Sorting Representations of relations: Matrices and Graphs Reflexive, Symmetric, Anti-Symmetric, and Transitive Equivalence Relations: using partitions and using functions Closures of Relations 6.5 Functions Functions Image and Inverse image Page 3 of 6

4 6.5.3 Injective, surjective and bijective functions Composition and inverse functions Recursive Definitions of Functions 6.6 Infinite Sets Equinumerous sets Countable and uncountable sets Cantor-Schroeder-Bernstein Theorem 6.7 Counting The Rules of Sum and Product The Pigeonhole Principle Permutations and Combinations The Binomial Theorem Combinations with Repetitions The Principle of Inclusion and Exclusion 6.8 Recursive Definitions of Discrete Structures and Mathematical Induction Recursively Defined Sequences Recursively Defined Sets: strings and formulas Induction: Complete and Structural 6.9 Proof techniques from the following topics Divisibility, Euclid s Algorithm and GCD Abstract Algebra, Linear Algebra, Real Analysis and Graph Theory 6.10 Additional Topics* Rook Polynomials Generating Functions Matching Theory Coding Theory Finite State Machines 7.0 Intended learning outcomes and associated assessment methods of those outcomes: Assessment Methods Learning Outcomes 7.1 Learn and recognize the usage of basic vocabulary, concepts, rules, definitions and standard logic needed for advanced mathematics Page 4 of 6

5 Assessment Methods Learning Outcomes 7.2 Solve challenging problems from various topics in advanced mathematics 7.3 Solve introductory problems from various topics in mathematics and apply them to problems in computer science 7.4 Write and explain coherent mathematical proofs 8.0 Program goals supported by this course: 8.1 To develop an understanding of the mathematical framework that supports engineering, science, and mathematics. 8.2 To develop critical and analytical thinking. 8.3 To develop an appropriate level of mathematical literacy and competency. 8.4 To provide an acquaintance with mathematical notation used to express physical and natural laws. 9.0 General education learning outcomes and/or goals supported by this course: Assessment Methods General Education Learning Outcomes 9.1 Communication Express themselves effectively in common college-level written forms using standard American English Revise and improve written and visual content Express themselves effectively in presentations, either in spoken standard American English or sign language (American Sign Language or English-based Signing) Comprehend information accessed through reading and discussion 9.2 Intellectual Inquiry Review, assess, and draw conclusions about hypotheses and theories Page 5 of 6

6 Assessment Methods General Education Learning Outcomes Analyze arguments, in relation to their premises, assumptions, contexts, and conclusions Construct logical and reasonable arguments that include anticipation of counterarguments Use relevant evidence gathered through accepted scholarly methods and properly acknowledge sources of information 9.3 Ethical, Social and Global Awareness Analyze similarities and differences in human experiences and consequent perspectives Examine connections among the world s populations Identify contemporary ethical questions and relevant stakeholder positions 9.4 Scientific, Mathematical and Technological Literacy Explain basic principles and concepts of one of the natural sciences Apply methods of scientific inquiry and problem solving to contemporary issues Comprehend and evaluate mathematical and statistical information Perform college-level mathematical operations on quantitative data Describe the potential and the limitations of technology Use appropriate technology to achieve desired outcomes 9.5 Creativity, Innovation and Artistic Literacy Demonstrate creative/innovative approaches to coursebased assignments or projects Interpret and evaluate artistic expression considering the cultural context in which it was created 10.0 Other relevant information: (such as special classroom, studio, or lab needs, special scheduling, media requirements, etc.) Smart classroom Page 6 of 6

Proof Theory for Syntacticians

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