Math 347. Fundamental Mathematics
Syllabus for Instructors

Text:
An Introduction to Abstract Mathematics, Donaldson, N. and A. Pantano. 2020. 

  • Introduction (1 day)
  • Chapter 2: Logic and Language of proofs (4–6 days)
    • Logical statements and connectives.
    • Methods of proof
    • Quantifiers; negating quantified statements
  • Chapter 3: Divisibility and congruence (2–3 days)
    • Optional:  Euclidean algorithm
    • Include discussion of congruence classes.
  • Chapter 4: Sets and functions (4–6 days)
    • Set notation, subsets, set operations
    • Proofs involving set containment/equality
    • Functions, composition of functions.
    • Injectivity, surjectivity, bijectivity.
    • The inverse of a bijective function (do not use the book’s treatment in terms of inverse relations).
  • Chapter 5: Induction (3–4 days)
    • Sum/product notation
    • Induction
    •  Strong induction
    •  the Well ordering principle
    • Equivalence of these three
    • Applications and examples.
  • Chapter 6: Set theory (2 days)
    • Cartesian products
    • Power sets
    • Arbitrary index sets.
  • Chapter 7: Relations and Equivalence relations (3–5 days)
    • Relations (inverse relations can be omitted if one introduces the inverse of a function earlier, which is recommended)
    • Equivalence relations
    • Partitions
    • Well definition of operations and functions on equivalence classes.
  • Chapter 8: Cardinality (4–6 days)
    • Basic definition of |A|=|B| as an equivalence relation
    • Definition of countable; cardinality of the integers and the rationals
    • Uncountable sets; cardinality of the reals
    • Schroder-Bernstein theorem (proof highly optional) and applications
    • Cardinality of power sets.
  • Additional topics (8–9 days)
    • A typical option is to discuss the real numbers and sequences; a reasonable free text is Lierl.  The handouts of A.J. Hildebrand are another good source.  A list of possible topics is below.
    • Completeness, supremum and infimum
    • Limits of sequences
    • The monotone convergence theorem
    • Limit laws
    • Cauchy sequences and subsequences
    • The Bolzano Weierstrass theorem and Cauchy’s criterion.
  • Exams, review, leeway (7 hours)