Chapter 0 – Logical Structures and Proofs

When I was studying Real Analysis for the first time, after graduating from an engineering program, I realized that I was always struggling to understand mathematical proofs.

After multiple (unsuccessful) attempts to learn real analysis on my own (usually by watching some online courses), I realized that the problem was not that “mathematical analysis” is hard, but rather that I had no experience with Logical Structures. Any undergraduate student in Math takes a course called Discrete Mathematics. During this course, they learn about Mathematical Logic, Proofs, Sets, and so on.

However, when someone (like myself) wants to venture in the world of Mathematical Analysis, they probably don’t know that they will have a hard time learning it without having gone through Discrete Math.

So, that’s the purpose of this chapter: to teach you some topics from Discrete Mathematics that I consider extremely important to avoid pitfalls when learning mathematical analysis.

You don’t need to master Set Theory, you don’t need to master Logic, you don’t need to master Discrete Mathematics. All you need is to learn the basics of transforming mathematical statements into logical structures and, after getting used to it, transforming this knowledge into what mathematicians like the most: intuition. (The rarely use logical forms to prove theorems, but learning about logical forms has opened my mind for understanding mathematical analysis).

Within Chapter 0, you’ll learn:

1. Basics of Logic
   • Sentential and Quantificational Logic (Truth Tables, Elements, Sets, Elementhood Test, Free and Bound Variables, Truth Sets, Universe of Discourse, Conditional and Biconditional Connectives, Universal and Existential Quantifiers, Logical Forms, Quantifiers Negation Laws, Vacuosly True Statements, Writing Elementhood Test Notation into Logical Forms)
2. Proof Strategies (without Quantifiers)
   • Hypothesis = Given and Conclusion = Goal
   • Proving a Goal
   • Proving a Goal of the Form “If P, then Q”
   • Proving a Goal of the Form “Contrapositive of If P, Then Q“
   • Proving a Goal of the Form “A Negative Statement” (by transforming into an equivalent positive statement)
   • Proving a Goal of the Form “A Negative Statement” (by using the Proof by Contradiction strategy – when it cannot be transformed into a positive statement)
   • Proving a Given (sparked by the use of Proof by Contradiction)
   • Proving a Given of the Form “If P, then Q” (Using the Rules of Inference: Modus Ponens and Modus Tollens)
3. Proof Strategies (with Quantifiers)
   • For all x / There Exists at Least One x
   • Proving a Goal of the Form “For All x, P(x) is True“
   • Proving a Goal of the Form “For All x, If P(x), Then Q(x)“
   • Proving a Goal of the Form “There Exists at Least one x (such that) P(x) is True“
   • Using a Given of the Form “There Exists at Least one x (such that) P(x) is True” (Using the Rule of Inference: Existential Instantiation)
   • Using a Given of the Form “For All x, P(x) is True” (Using the Rule of Inference: Universal Instantiation)
4. Proof Strategies for Conjunctions (P and Q)
5. Proof Strategies for Disjunctions (P or Q)
6. Proof Strategies for Biconditionals (P If and only If Q)
7. Proof Strategies: Existence and Uniqueness
8. Relations
9. Sentential and Quantificational Logic
10. Function
11. Mathematical Induction
12. Infinite Sets