Syllabus Math 347

 

Math 347. Fundamental Mathematics
Syllabus for Instructors

Text:
D'Angelo/West, Mathematical Thinking, Second Edition, ISBN 0-13-014412-6

  • Chapter 1: Numbers, Sets, and Functions (2 - 4 hours)
    • Set-theoretic concepts and operations
    • Functions: Formal definition and basic concepts (graph, image set, bounded function, decreasing/increasing function)
    • Triangle and AGM inequalities
    Note: Much of the material in Chapter 1 can be covered at a later point. For example, one could spend one or two hours on basic set-theoretic concepts, then move on to logical statements in Chapter 2, while deferring the formal definition of functions and related concepts to the beginning of Chapter 4.
     
  • Chapter 2: Language and Proof (4 - 6 hours)
    • Logical statements, conditionals, quantifiers
    • Methods of proof (direct, contraposition, contradiction)
  • Chapter 3: Induction (4 - 6 hours)
    • Sum/product notations
    • Induction and strong induction
    • Well-ordering principle
    • Applications
  • Chapter 4: Bijections and Cardinality (4 - 6 hours)
    • More about functions: injective, surjective, bijective properties, compositions and inverses
    • Cardinality, finite, countable, and uncountable sets
    • Countability of rationals
  • Chapters 13/14: Real Numbers, Sequences and Series (8 - 10 hours)
    • Infinite sequences and series: Formal definition of convergence, basic properties of convergent sequences and series
    • Sup, inf, and the Completeness Axiom
    • Cauchy sequences and the Cauchy Convergence Criterion
    • Montone Convergence Theorem
    • Bolzano-Weierstrass Theorem
    Note: These chapters introduce students to basic concepts in analysis and form an essential component of this course. Most students find this part to be the most difficult of the course, so it is important not to rush through it, and it would be good to cover Chapters 13 and 14 before any additional topics.
     
  • Additional Topics (8 - 12 hours)
    Selected topics chosen from the remainder of the text. Possible choices include Number Theory (Chapters 6 and 7), Combinatorics (Chapter 5, 10 - 12), and Probability (Chapter 9).
     
  • Leeway and Hour Exams (6 hours)