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

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1 ! ROCHESTER INSTITUTE OF TECHNOLOGY COURSE OUTLINE FORM COLLEGE OF SCIENCE School of Mathematical Sciences New Revised COURSE: COS-MATH-200 Discrete Mathematics and Introduction to Proofs 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: Discrete Mathematics and Introduction to Proofs Credit Hours: 3 Prerequisite(s): COS-MATH-182 or equivalent 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: 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) First-year Applied Mathematics, Computational Mathematics and Applied Statistics majors Students who might elect to take the course: Students majoring in Physics, Imaging Science, Computer Science, Bio-informatics, Computer Engineering, and Software Engineering, and students 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 develop mathematical maturity and change perspective from problem solving to theorem proving. 3.2 To learn how to construct basic proofs, how to read mathematics, and how to write mathematics. 3.3 To learn various methods of mathematical proof with emphasis on applications, and illustrations of these methods via topics in the three main pillars of mathematics: algebra, analysis, and geometry. 3.4 To develop skills in concise exposition, cogent communication of mathematical ideas and the presentation of proofs. 4.0 Course description: (as it will appear in the RIT Catalog, including pre- and co-requisites, semesters offered) COS-MATH-200 Discrete Mathematics and Introduction to Proofs This course provides preparation for professions that use mathematics in daily practice or university mathematics courses beyond the introductory level where it is essential to communicate effectively in the language of mathematics. It covers various methods of mathematical proof, starting with basic techniques in propositional and predicate calculus and set theory, and then moving to applications in advanced mathematics. (Pre-requisite(s): COS- MATH-182, or COS-MATH-173) 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, MA. 5.2 R. Garnier and J. Taylor, 100% Mathematical Proof, Wiley, Hoboken, NJ. 5.3 D. J. Velleman, How to Prove It: A Structured Approach, Cambridge University Press, Cambridge, UK. 5.4 J. P. D Angelo and D. B. West, Mathematical Thinking: Problem-Solving and Proofs, Addison-Wesley, Reading, MA. 5.5 G. Chartrand, A. D. Polimeni and P. Zhang, Mathematical Proofs: A Transition to Advanced Mathematics, Addison-Wesley, Reading, MA. 5.6 T. G. Faticoni, The Mathematics of Infinity, Wiley, Hoboken, NJ. Page 2 of 6

3 6.0 Topics: (outline) Topics with an asterisk(*) are at the instructor s discretion, as time permits 6.1 Informal 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 Sets and operations on sets Indexed families of sets Venn diagrams Examples of sets: R, Q, Z, C 6.4 Relations Relations as cartesian product Ordering relations Equivalence relations and partitions Partially ordered sets 6.5 Functions Functions Image and inverse image Injective, surjective and bijective functions Composition and inverse functions 6.6 Infinite Sets Equinumerous sets Countable and uncountable sets Cantor-Schroeder-Bernstein theorem 6.7 Structure of N Well-ordering principle No infinite descending sequences Mathematical induction 6.8 Proof techniques from the following topics Counting: permutations and combinations Page 3 of 6

4 6.8.2 Divisibility, Euclid s algorithm and the greatest common divisor Iterations of functions Abstract algebra, linear algebra, real analysis and graph theory 7.0 Intended learning outcomes and associated assessment methods of those outcomes: Assessment Methods Learning Outcomes 7.1 Identify and apply basic vocabulary, concepts, rules, definitions and standard logic needed for advanced mathematics 7.2 Solve the standard and advanced problems in set theory, relations, functions, equivalence relations and partial orders 7.3 Write and explain mathematical proofs 7.4 Solve introductory problems from various topics in advanced mathematics Homework Quiz/Exam/Final Project Computer Work Class Presentation 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. 8.5 To produce graduates who can effectively use mathematics and/or statistics to model, analyze, and solve problems arising in science, engineering, business, and other disciplines. 9.0 General education learning outcomes and/or goals supported by this course: Page 4 of 6

5 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 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 Homework Quiz/Exam/Final Project Computer Work Class Presentation Page 5 of 6

6 Assessment Methods General Education Learning Outcomes Demonstrate creative/innovative approaches to coursebased assignments or projects Interpret and evaluate artistic expression considering the cultural context in which it was created Homework Quiz/Exam/Final Project Computer Work Class Presentation 10.0 Other relevant information: (such as special classroom, studio, or lab needs, special scheduling, media requirements, etc.) None Page 6 of 6