MAT 271 Foundations of Higher Mathematics
This is a sample syllabus only. Ask your instructor for the
official syllabus for your course.
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Expanded Course Description
Prepares students for the transition from lower division
mathematics courses  which are often based on computation  to
upper division mathematics courses  which 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.
MAT 271 meets for three hours of lecture per week.
Prerequisites
Required: MAT 191 with grade C or better.
Objectives
After completing MAT 271 the student should be able to
 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 counterexamples
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
Expected outcomes
Students should be able to demonstrate through written
assignments, tests, and/or oral presentations, that they
have achieved the objectives of MAT 271.
Method of Evaluating Outcomes
Evaluations are based on homework, class participation,
short tests and scheduled examinations covering students'
understanding of topics covered in MAT 271.
Text
A Transition to Advanced Mathematics (4th ed.), by
Douglas Smith, Maurice Eggen, Richard St. Andre. rooks/Cole
Publ., 1997. (Chapters 15).
Table of contents
 Chapter 1 Logic and Proofs

 1.1 Propositions and Connectives
 1.2 Conditionals and Biconditionals
 1,3 Quantifiers
 1.4 Mathematical Proofs
 1.5 Proofs Involving Quantifiers
 1.6 Additional Examples of Proofs
 Chapter 2 Set Theory

 2.1 Basic Notations of Set Theory
 2.2 Set Operations
 2.3 Extended Set Operations and Indexed Families of
Sets
 2.4 Induction
 2,5 Equivalent Forms of Induction
 2.6 Principle of Counting (if time allows)
 Chapter 3 Relations

 3.1 Cartesian Products and Relations
 3.2 Equivalence Relations
 3.3 Partitions
 3.4 Ordering Relations
 3.5 Graphs of Relations
 Chapter 4 Functions

 4.1 Functions as Relations
 4.2 Constructions of Functions
 4.3 Functions That Are Onto; OnetoOne
Functions
 4.4 Induced Set Functions Chapter 5 Cardinality
 5.1 Equivalent Sets; Finite Sets
 5.2 Infinite Sets
 5.3 Countable Sets
 5.4 The Ordering of Cardinal Numbers
 5.5 Comparability of Cardinal Numbers and the Axiom
of Choice
 Chapter 8 Euclidean Spaces

 8.1 Arrows and Vectors in Intrinsic Euclidean
Geometry
 8.2 Addition and Scalar Multiplication of
Vectors
 8.3 Identification of Vectors with Points of R^n
 8.4 Properties of Vectors
 8.5 Dot Product
 8.6 Cross Product
 Chapter 9 Complex Numbers

 9.1 Definition of Complex Numbers (following
Gauss)
 9.2 Absolute Value, Argand Form, de Moivre
Formula
 9.3 nth Root of Complex Numbers
 9.4 Fundamental Theorem of Algebra
Grading Policy
Students' grades are based on homework, class participation,
short tests, and scheduled examinations covering students'
understanding of the topics covered in MAT 271. The instructor
determines the relative weights of these factors. During tests
students may be allowed a sheet of formulas (written by the
student or provided by the instructor) and a graphing
calculator as appropriate at the instructor's discretion.
Attendance Requirements
Attendance policy is set by the instructor.
Policy on Due Dates and MakeUp Work
Due dates and policy regarding makeup work are set by
the instructor.
Schedule of Examinations
The instructor sets all test dates except the date of the
final exam. The final exam is given at the date and time
announced in the Schedule of Classes.
Academic Integrity
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 I can best help you. All disclosures of disabilities will be kept strictly confidential. Please note: no accommodation may be made until you register with the DSS in WH B250. For information call (310) 2433660 or to use telecommunications Device for the Deaf, call (310) 2432028.
Revision history:
Prepared by F. Brulois 9/22/00. Revised 7/7/01, 7/25/06, 1/21/11 (G. Jennings).