Taylor J., Garnier R. Understanding Mathematical Proof 2014

Textbook in PDF format

The need for proof.
The language of mathematics.
Reasoning.
Deductive reasoning and truth.
Example proofs.
Logic and Reasoning.
Propositions, connectives, and truth tables.
Logical equivalence and logical implication.
Predicates and quantification.
Logical reasoning.
Sets and Functions.
Sets and membership.
Operations on sets.
The Cartesian product.
Functions and composite functions.
Properties of functions.
The Structure of Mathematical Proofs.
Some proofs dissected.
An informal framework for proofs.
Direct proof.
A more formal framework.
Finding Proofs.
Direct proof route maps.
Examples from sets and functions.
Examples from algebra.
Examples from analysis.
Direct Proof: Variations.
Introduction Proof using the contrapositive.
Proof of biconditional statements.
Proof of conjunctions.
Proof by contradiction.
Further examples.
Existence and Uniqueness.
Constructive existence proofs.
Non-constructive existence proofs.
Counter-examples.
Uniqueness proofs.
Mathematical Induction.
Proof by induction.
Variations on proof by induction.
Hints and Solutions to Selected Exercises