Introduction to Advanced Mathematics: How to Solve Proofs Book

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.