Mathematical Proof

Sometimes people read mathematical proofs and think they are reading a foreign language. This book describes the language used in a mathematical proof and also the different types of proofs used in math. This knowledge is essential to develop rigorous mathematics. As such, rigorous knowledge of math is not a prerequisite to reading this book. This book will use some set and logic notations for communication, and you should be familiar with those notations after learning more about set theory and logic in the first two chapters.

After introducing set theory informally (i.e. not emphasizing on axioms in set theory) [1] and logic, we will be prepared to study methods of mathematical proof. After that, we will be discussing some fundamental concepts that are important for more advanced topics in mathematics.

A Venn diagram

Table of ContentsEdit

  1. Introduction to Set Theory
  2. Logic  
  3. Methods of Proof  
    1. Direct Proof  
    2. Proof by Contrapositive  
    3. Proof by Contradiction  
    4. Proof by Induction  
    5. Counterexamples  
    6. Other Proof Types  
    7. Proof and Computer Programs (optional)  
    8. Proof Assistants (optional)  
  4. Equivalence Relations
  5. Functions
  6. Set Cardinalities


  1. G. Chartrand, A.D. Polimeni and P. Zhang (2018). "Chapter 0-12". Mathematical Proofs: A Transition to Advanced Mathematics (4th ed.). Pearson. ISBN 9780134746753. 
  1. We introduce set theory informally since this is simpler and set theory is not the main focus of this book. For a formal discussion of set theory (which may be difficult to understand without the knowledge learnt in this book), see the wikibook Set Theory. Even if the introduction is informal, it will still be clear and useful.