Math 290: Fundamentals of Mathematics.

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Catalog Information

Title

Fundamentals of Mathematics.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

F, W, Sp

Prerequisite

Math 112 or concurrent enrollment with instructor's consent.

Description

Achieving maturity in mathematical communication. Introduction to mathematical proof; methods of proof; analysis of proof; induction; logical reasoning.

Desired Learning Outcomes

This course is aimed at undergraduate mathematics and mathematics education majors. It is a first course in mathematical thinking. It is intended as an introduction to mathematical proof, and students who finish the course should achieve maturity in mathematical communication.

Prerequisites

This course has no prerequisites.

Minimal learning outcomes

Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the below concepts, related to, but not identical to, statements proven by the text or instructor.

  1. Set Theory
    • Set builder notation
    • Venn diagrams
    • De Morgan’s Laws
  2. Logic
    • Truth Tables
    • Quantifiers
    • Negations of statements with quantifiers
    • Implications
    • Biconditionals
  3. Proof Techniques
    • Direct proof
    • Proof by contrapositive
    • Proof by contradiction
  4. Relations
    • Reflexive, irreflexive, symmetric, transitive relations
    • Equivalence relations
    • Equivalence classes
  5. Functions
    • One-to-one and onto
    • Function composition
    • Inverse functions
    • Bijective functions
    • Permutations
  6. Mathematical Induction
    • Well ordering principle
    • Mathematical induction
    • Strong induction
    • The method of descent
  7. Cardinal Numbers
    • Numerical equivalence
    • Countable and uncountable sets
    • Schröder-Bernstein theorem
  8. Number Theory
    • Division algorithm
    • Euclid’s Algorithm
    • Infinitude of primes
    • Unique factorization

In addition, on completion of the course, students should understand the basic mathematical language concerning logic, sets, the standard number systems, deductive and inductive reasoning, and the structure of proof. They should be able to translate a mathematical statement into logical form and discuss its negation and its implications. They should be able to translate a simple argument into logical form and detect logical validity and flaws. They should be able to read, write, listen and speak using standard mathematical terminology and reasoning.

Textbooks

Possible textbooks for this course include (but are not limited to):

  • Gary Chartrand, Albert D. Polimeni, and Ping Zhang, Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition), Addison Wesley, 2007.


Additional topics

Beyond the minimal learning outcomes, instructors are free to cover additional topics. These may include (but are certainly not limited to): concepts of set theory, number theory, geometry, analysis, group theory, and ring theory. Instructors are free to use new approaches to the teaching of the material, as long as the core topics are adequately covered.

Courses for which this course is prerequisite

This course is required for almost all upper division course in the Mathematics department. There is a strong expectation that students who have taken this course will have a certain minimal preparation in mathematical thinking. As a result it is essential that all required learning outcomes be thoroughly covered.