Proofs
and
Concepts
the fundamentals
of abstract mathematics
by Dave Witte Morris and Joy Morris
with thanks to P D Magnus
This free undergraduate textbook provides an introduction to proofs,
logic, sets, functions, and other fundamental topics of abstract
mathematics. It is designed to be the textbook for a bridge
course that introduces undergraduates to abstract mathematics, but it
is also suitable for independent study by undergraduates (or
mathematically mature high-school students), or for use as a very
inexpensive supplement to undergraduate courses in any field of
abstract mathematics.
The book is being released online with a Creative Commons
license (Attribution-NonCommercial-ShareAlike 2.0). Although not
in final form, it has already been
used as a textbook for 3 semesters at the University of
Lethbridge. The preliminary version posted in May 2009 is
approximately 220
pages.
Click
here for a PDF file of the most recent version (approx 1.8MB).
Table of Contents
Part I. Introduction to Logic and Proofs
1. What is Logic?
2. Propositional Logic
3. Basic Theorems of Propositional Logic
4. Two-Column Proofs
Part II. Sets and First-Order Logic
5. Sets, Subsets, and Predicates
6. Operations on Sets
7. First-Order Logic
8. Quantifier Proofs
Part III. Functions
9. Functions
10. One-to-one Functions
11. Onto Functions
12. Bijections
13. Inverse Functions
14. Composition of Functions
Part IV. Other Fundamental Concepts
15. Cardinality
16. Proof by Induction
17. Divisibility and Congruence
18. Equivalence Relations
Part V. Topics
19. Elementary Graph Theory
20. Isomorphisms