--- title: Proof Techniques description: Proof techniques are formal methods used to establish the mathematical truth of statements. You will understand how to construct rigorous arguments using direct proof, contradiction, contraposition, mathematical induction, and case analysis to solve diverse mathematical problems. category: mathematics subcategory: mathematical-logic difficulty: beginner, intermediate, advanced url: /subject/proof-techniques --- # Proof Techniques Proof techniques are formal methods used to establish the mathematical truth of statements. You will understand how to construct rigorous arguments using direct proof, contradiction, contraposition, mathematical induction, and case analysis to solve diverse mathematical problems. ## Available Resources 4 Books • 2 Courses • 2 Websites ## Courses ### 1. Mathematics for Computer Science (MIT 6.042J) **Author:** Tom Leighton, Marten van Dijk Discrete mathematics for computer science with an emphasis on definitions and proofs: logic, induction, sets and relations, graph theory, modular arithmetic, asymptotics, counting and discrete probability. 25 lecture videos, problem sets and exams with solutions build fluency in writing proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/ **Tags:** discrete-mathematics, mathematical-proofs, induction, graph-theory, combinatorics, discrete-probability ### 2. Introduction to Mathematical Thinking **Author:** Keith Devlin Stanford course by Keith Devlin on how mathematicians reason, starting with the logic of language (and, or, not, implication, quantifiers) and moving to the structure of proofs, including contradiction and induction. Learners become able to read, construct and critique simple mathematical proofs. **Difficulty:** Beginner | **Language:** English | **Duration:** Expect to require at least 10 hours of study per week to complete this course satisfactorily. | **Price:** Free **Link:** https://www.coursera.org/learn/mathematical-thinking **Tags:** mathematical-logic, proof-writing, quantifiers, mathematical-reasoning ## Books ### 1. How to Solve It: A New Aspect of Mathematical Method **Author:** George Pólya George Pólya's classic guide to mathematical problem solving, built around a four-step method of understanding, planning, carrying out and looking back, plus a dictionary of heuristics such as working backwards, auxiliary problems and generalization. Readers learn to attack unfamiliar problems and proofs systematically. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X?tag=edmonddante07-20 **Tags:** problem-solving, heuristics, mathematical-reasoning, proof-strategies ### 2. How to Read and Do Proofs - Daniel Solow **Author:** Daniel Solow Introduces the "Forward-Backward" method to help students understand the structure and logic of mathematical proofs. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://amzn.to/4bpzo6V **Tags:** proof-techniques, mathematical-logic, forward-backward-method, undergraduate-mathematics, textbook ### 3. How to Prove It: A Structured Approach **Author:** Daniel J. Velleman Velleman builds proof-writing from propositional and predicate logic upward, then works through set theory, relations, functions, and induction. Exercises demand full written proofs. Standard bridge text for students moving from calculus to abstract mathematics. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0521446635?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematical-logic ### 4. Mathematical Proofs: A Transition to Advanced Mathematics Mathematical Proofs: A Transition to Advanced Mathematics - A book resource **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0201710900?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, mathematical-logic ## Websites ### 1. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ### 2. proofwiki.org ProofWiki is a free, collaboratively edited online collection of mathematical proofs, theorems, and definitions organized by technique. It provides step-by-step proofs and explanations of proof strategies (such as induction, contradiction, and contrapositive) across a wide range of mathematical topics. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://proofwiki.org **Tags:** websites, mathematics-statistics, mathematical-logic --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/proof-techniques