# MAT 271 Foundations of Higher Mathematics

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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 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

### 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 1-5).

• 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; One-to-One 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

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 Make-Up Work

Due dates and policy regarding make-up 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.