Dave4Math / Courses / Logic and Proof

Logic and Proof

Master logic and proofs. With these lessons and videos, you'll learn about different styles and techniques of mathematical proofs. This course takes a beginner and introduces them to logic and then demonstrates (through discourse) how to write proofs.


January 11, 2021

Created by Dave

This course begins Summer 2021.
Current Status
Not Enrolled
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Have you ever been excited about going to a math class? The class was so organized, and compelling, and the subject matter was taught in a very clear way.  Introducing Dave’s online math course: Logic and Proof. You’ll not only be knowledgeable and skilled, but enjoy it as well. 

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in logic and writing mathematical proof.
  • Students who are enrolled in Introduction to Proofs and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about propositional logic, quantifiers, mathematical proofs, and logical discourse.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

What You’ll Learn in Logic and Proof

  • Mathematical Statements
  • Logical Connectives
  • Constructing Truth Tables
  • Tautologies and Contradictions
  • Contrapositive, Converse, and Inverse
  • Modus Ponens and Substitution
  • Logical Equivalence
  • Introduction to Quantifiers
  • Propositional Functions
  • Universal Quantifier
  • Existential Quantifier
  • Uniqueness Quantifier
  • Negating Quantifiers
  • Counterexamples
  • Valid Arguments
  • Combining Quantifiers
  • Inference Rules
  • Direct Proofs
  • Indirect Proofs
  • Proof by Contrapositive
  • Proof by Cases
  • Simple Examples
  • Axiomatic Systems
  • Inference Rules for Quantified Statements
  • Theorems in Incidence Geometry
  • Summary of Logical Discourse

Course Description

We begin this course on logic and proof by discussing what mathematical statements are. We start with an elementary study of propositional logic by constructing truth tables and understanding their properties. We discuss tautologies, contradictions, and contingencies and also talk about contrapositives, converses and inverses.

The second part of this course discusses quantifiers. We present various types of quantifiers and study negating quantifiers. We pay special attention to combining quantifiers and writing valid arguments by working through several examples.

Writing mathematical arguments is the emphasis of this course. We examine two main strategies for writing proofs: a column proof versus a paragraph proof. We illustrate both approaches by considering various types of proof such as direct proof, indirect proof, proof by cases, and proof by contrapositive.

We finish this course with a discussion of how to perform logical discourse. Axiomatic systems are discussed in great detail, including an example of an axiomatic system called Incidence Geometry. We will prove several fundamental results using both strategies discussed above.

I recommend the prerequisite course Sequences, Series, and Probability. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free articles on introduction to proofs.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.