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.

Table of Contents:

Part I. Introduction to Logic and Proofs

• What is Logic?
• Propositional Logic
• Two-Column Proofs

Part II. Sets and First-Order Logic

• Sets, Subsets, and Predicates
• Operations on Sets
• First-Order Logic
• Quantifier Proofs
• Divisibility and Congruence

Part III. Other Fundamental Concepts

• Functions
• Cardinality
• Proof by Induction
• Equivalence Relations