Math 300

Introduction to Mathematical Reasoning

Math 300 is a course emphasizing mathematical arguments and the writing of proofs. The course gives students the opportunity to learn how to formulate mathematical arguments in an elementary mathematical setting. It serves as a complement to calculus by introducing ideas of discrete mathematics. In addition to forming a foundation for more abstract mathematics it should appeal to students preparing to be teachers or computer scientists.

The topics covered will include:


1. Logic and Mathematical Language

  • Compound statements (``and,'' ``or,'' ``implies'') and their negations.
  • The meaning of ``necessary,'' ``sufficient,'' ``if,'' ``only if,'' contrapositive, converse.
  • Quantified statements and their negations.

    • 2. Proofs

  • Direct, indirect, and contrapositive proofs.
  • Existence and uniqueness proofs.
  • Disproof by counterexample.
  • Proof by induction.

    • 3. Elementary Set Theory

  • Basic ways of defining sets.
  • Subsets, unions, intersections, set differences.
  • How to prove one set is a subset of another, or equal to another.
  • Finite Cartesian products.

    • 4. Functions

  • Definition of a function between two sets, and what it means for a function to be well-defined.
  • What it means for two functions to be equal.
  • Injectivity, surjectivity, bijectivity.
  • Composition of functions.
  • Inverse functions.

    • 5. Cardinality

  • What it means for two sets to have the same cardinality.
  • What it means for a set to be finite, infinite, countable, uncountable, and how to prove it.
  • Subsets, finite products, and finite unions of finite sets are finite.
  • Subsets, finite products, and finite unions of countably infinite sets are at most countable.
  • The rationals are countable, the reals are not.

    • Aside from these required topics, the course content and the choice of textbook are up to the instructor. For example, an instructor might focus on arithmetic (divisibility, prime numbers, modular arithmetic); the real numbers (absolute values, square roots, solving equations, inequalities, sequences, increasing and decreasing functions); set theory (equivalence relations, order relations, axiom of choice, Russell paradox); probability (finite probabilities, counting arguments, expectation); combinatorics (binomial coefficients, enumeration, graph theory); or something else.