University of Rhode Island MTH307: Introduction to Mathematical Rigor
Spring 2016


Here is a basic outline of the topics we will cover (time permitting!) this semester.
  1. Fundamental Concepts
    • Sets: Describing sets, products, unions, intersections and complements, Venn diagrams.
    • Logic: Statements, compounds, conditional statements, quantifiers, negations.
    • Counting: (optional) Lists, factorials, combinatorics, the inclusion-exclusion principle.
  2. Proving conditional statements
    • Direct proof: Theorems and definitions, cases.
    • Contrapositives: Proofs and exposition.
    • Proof by contradiction: Proof and combining techniques in a proof.
  3. More Proofs
    • Non-conditional statements: Equivalences, existence and uniqueness.
    • Proofs and sets: Set inclusion, subsets, equality.
    • Disproof: Counterexamples and contradiction.
    • Induction: Mathematical induction, examples.
  4. Relations, functions and cardinality
    • Relations: Relations, equivalence relations, partitions.
    • Functions: Injectivity, surjectivity, compositions, inverses, images.
    • Cardinality: Equal cardinality, countability, Cantor-Schroder-Bernstein Theorem
    • .