This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed.

### Table of Contents

1. LOGIC.

Introduction and Terminology. Statements and Truth Tables. Logical Equivalence and Logical Deductions. The Contrapositive, Negation, and Converse of an Implication Statement. Quantifiers.

2. PROOF WRITING.

Terminology and Goals. Existence Proofs and Counterexamples. Direct Proofs ("If, then" or "For every" Statements). Using Cases in Proofs. Contrapositive Arguments. Contradiction Arguments. Putting it All Together. Regular Induction. Induction with Inequalities. Recursion and Extended Induction. Uniqueness Proofs, the WOP, and a Proof of the Division. Algorithm.

3. SETS, RELATIONS, AND FUNCTIONS.

Sets. Set Operations. Set Theory. Indexed Families of Sets. Cartesian Products. Relations. Functions. Composition of Functions. Cardinality.