--- title: Relations description: Relations define connections between elements of sets, forming the basis for orderings and equivalence. Learners will understand properties like reflexivity, symmetry, and transitivity, and be able to analyze equivalence relations, equivalence classes, and partial orders. category: mathematics subcategory: set-theory difficulty: beginner, intermediate, advanced url: /subject/relations --- # Relations Relations define connections between elements of sets, forming the basis for orderings and equivalence. Learners will understand properties like reflexivity, symmetry, and transitivity, and be able to analyze equivalence relations, equivalence classes, and partial orders. ## Where to start Start with Mathematics for Computer Science (MIT 6.042J) by Tom Leighton and Marten van Dijk, a free MIT OpenCourseWare course that teaches sets and relations alongside logic and proof, with lecture videos and problem sets with solutions. If you only use one resource, stay with this course. For the set-theoretic foundations, continue with Herbert B. Enderton's Elements of Set Theory. ## Available Resources 1 Books • 2 Courses • 3 Websites ## Websites ### 1. PlanetMath - Binary Relation Encyclopedia entry on binary relations in set theory **Difficulty:** Beginner | **Price:** Free **Link:** https://planetmath.org/binaryrelation **Tags:** binary-relations, set-theory, discrete-mathematics ### 2. ProofWiki - Relations Category index on ProofWiki collecting formal definitions and fully written proofs about relations: reflexivity, symmetry, transitivity, equivalence relations, orderings, composition and inverses. Useful for checking a precise statement or seeing how a standard result is proved step by step. **Difficulty:** Beginner | **Price:** Free **Link:** https://proofwiki.org/wiki/Category:Relations **Tags:** binary-relations, equivalence-relations, order-relations, proofs, set-theory ### 3. 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 ## 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. Elements of Set Theory **Author:** Herbert B. Enderton Undergraduate textbook developing axiomatic Zermelo-Fraenkel set theory from the ground up. Covers relations and functions, the construction of the natural and real numbers, cardinals, ordinals and the axiom of choice, leaving the reader able to work with sets rigorously. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0122384407?tag=edmonddante07-20 **Tags:** set theory, zfc, relations and functions, cardinal numbers, ordinal numbers, axiom of choice --- *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/relations