# MAT 271 Foundations of Higher Mathematics

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### Course Description

The official course description in the university catalog reads as follows "Topics include logic, methods of mathematical proof, set theory, relations and functions. Introduction to complex numbers and proof strategies using ideas of vector algebra. Meant to prepare students for mathematics program as well as concepts of computer science."

3 units credit.

Here is an unofficial, expanded description of MAT 271. "Prepares students for the transition from lower division mathematics courses - which are often based on computation - to upper division mathematics courses that typically are based on proof. Mathematical rigor, proof strategies, and writing are emphasized. Covers elementary mathematical logic, including propositional and predicate calculus, set theory, equivalence and order relations, simple and directed graphs, functions, and cardinals. Presents a rigorous treatment of vectors in Euclidean space and complex numbers as illustrative examples."

### Prerequisite

Required: MAT 191 with grade C or better.

### Text

Texts are chosen by the instructor. For example:

Book of Proof (2nd ed.), by Richard Hammack. Free download at http://www.people.vcu.edu/~rhammack/BookOfProof

### Course Requirements, Tentative Schedule of Class Meetings and Topics, Readings, Assignments and Due dates, Exams

A schedule of class meetings, topics, assignments, due dates, exam dates, etc. will be provided by instructor. See your class syllabus.

Here is an example course, based on the above text. Other topics might be included as time permits.

• Part I: Fundamentals
• 1. Sets
• 2. Logic
• 3. Counting
• Part II: How to Prove Conditional Statement
• 4. Direct Proof
• 5. Contrapositive Proof
• Part III: More on Proof
• 7. Proving Non-Conditional Statements
• 8. Proofs Involving Sets
• 9. Disproof
• 10. Mathematical Induction
• Part IV: Relations, Functions and Cardinality
• 11. Relations
• 12. Functions
• 13. Cardinality

The final exam is given at the date and time announced in the Schedule of Classes.

### Learning Objectives

After completing MAT 271 students will

• critique a purported proof
• use a variety of proof strategies in proving propositions, including direct proof, proof by contraposition, proof by contradiction, proof by exhaustion, proof by induction
• devise existence proofs, either constructive or using other existential proposition
• devise uniqueness proofs and understand the need for such
• prove economically that two or more statements are equivalent
• write proofs that are logically coherent, written in grammatically correct English, using standard mathematical ideas in undergraduate mathematics courses and textbooks
• understand the concept of, and construct counter-examples to, false statements
• produce truth tables for statements in the propositional calculus
• negate compound and quantified propositions
• use reliably the concepts of elementary set theory, including set notation, set operations, inclusion, subsets, power sets, indexed families of sets and their union and intersection, Cartesian product, binary relations including equivalence and order relations, partitions and their connection to equivalence relations, simple and directed graphs, equivalent sets, cardinals, finite sets, countable sets
• operate in a formal and rigorous way with the concept of function and related concepts, including composition of functions, inverse of a function, restriction of a function, injections, surjections, and bijections, induced set functions
• perform standard vector computations, including sum, scalar multiplication, length, dot and cross product, projection of vector onto another
• perform standard complex number computations, including sum, difference, product, and quotient of complex numbers, roots of complex numbers
• find the zeroes of real polynomials including multiplicity and conjugate pairs throughout, use standard mathematical notation and terminology and avoid nonsensical expressions and statements

### Computers and Calculators, Computer Literacy

Most instructors encourage the use of machines, calculators computers, phones etc., for analyzing data. The use of machines may be restricted during examinations or at certain other times. Ask your instructor for the policy in your class.

Students are not expected to be programmers or to know any particular computer language before starting this class. Some instructors may expect students to be able to access information on the internet, or to use calculators, or to learn to use particular software with instruction. Basic skill in algebra and the use of mathematical symbols, order of operations etc., and the willingness to read and follow instruction manuals and help files will suffice.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in this course. The instructor determines the relative weights of these factors and the grading scale. See the syllabus for your particular class.

### Location of Class Meetings

Classes meet on the dates and room announced in the official Schedule of Classes. This is a traditional, face-to-face class.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates, Make-Up Work, Missed Exams, and Extra-Credit Assignments

Due dates and policy regarding make-up work and missed exams are set by the instructor. Instructors may, or may not, choose to offer extra credit assignments. If extra credit assignments are offered they will be available to all students.

The mathematics department does not tolerate cheating. Students who have questions or concerns about academic integrity should ask their professors or the counselors in the Student Development Office, or refer to the University Catalog for more information. (Look in the index under "academic integrity".)

### Accomodations for Students with Disabilities

Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how we best can help you. All disclosures of disabilities will be kept strictly confidential. Please note: you must register with DSS to arrange an no accommodation. For information call (310) 243-3660 or send an email message to dss@csudh.edu or visit the DSS website http://www4.csudh.edu/dss/contact-us/index or visit their office WH D-180

### Behavioral Expectations

We all are adults so behavior rarely is an issue. Just follow the Golden Rule: "do unto others as you would have them do unto you" then everything will be fine.

The university must maintain a classroom environment that is suitable for learning, so anyone who insists on disrupting that environment will be expelled from the class.

Revision history:

Prepared by F. Brulois 9/22/00. Revised 7/7/01, 7/25/06, 1/21/11, 1/10/16 (G. Jennings).