Introduction to abstract mathematics: course

Book

Description

This is a one-semester course to introduce math majors into pure/abstract mathematics. It's mostly about proofs. However, I would like to introduce a few other things that will make linear algebra, advanced calculus, group theory, topology, etc easier to absorb later on.

Aka:

• Introduction to higher mathematics
• Introduction to advanced mathematics
• Introduction to proofs

Prerequisites

• Calculus 1-3

Outline

About proofs...

"Because I don’t have time to write you a short letter, I write you a long one." Attributed to Hemingway, Cicero, Voltaire, Mark Twain, Blaise Pascal, Goethe.

In writing proof we'd better follow the latter approach, especially when we are still learning.

"If you like laws and sausages, you should never watch either one being made." (Mis)attributed to Bismarck.

We'll have to watch, if we want to learn how to do it.

In addition to proofs we need to start to become familiar with various mathematical constructions. They should be illustrated with examples from previous courses (calculus and precalculus).

• Rules of proof writing
• Introduction:
  • Sloppy notation.
  • examples and types of proofs:
    • computation,
    • contradiction,
    • induction,
    • counterexamples.
• Proofs:
  • reading proofs,
  • checking proofs,
  • discovering proofs,
  • writing proofs.
• Trivialization as a way of understanding concepts, proofs etc
• Examples:
  • Proofs in Euclidean geometry
  • Proofs about numbers
  • Proofs in calculus
• Abstract mathematics, axioms, definitions, and well-defined concepts:
  • turning theorems into definitions,
  • axioms of Euclidean geometry,
  • axioms of topology,
  • coordinates,
  • limit,
  • Riemann integral,
  • PageRank,
  • circumnavigation
  • definition-theorem-proof.
• Sets:
  • subsets,
  • the power set,
  • unions, intersections, complements
• Functions:
  • identity function, constant functions,
  • compositions,
  • commutative diagrams:
    • algebra of limits,
  • one-to-one and onto,
  • bijections, inverses,
  • images and preimages,
  • inclusions, restrictions, extensions.
• New sets from old:
  • product sets and projections
  • equivalence relations, partitions,
  • quotient sets, the quotient function:
    • antiderivatives,
    • modular arithmetic
• New sets and functions from old:
  • function-valued functions,
  • spaces of functions,
  • dual spaces, duality

Review

• Is a set a subset of itself?
• Is projection a quotient function?

Notes

Text:

Further reading

• Linear algebra: course
• Group theory: course