binary relation
Let an agent explain binary relations and their formal definition and properties, describe important kinds and their uses, support proofs and exercises, and support learning in mathematics and computing.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify
Researched by: Claude
Purpose and description
Let an agent explain binary relations and their formal definition and properties, describe important kinds and their uses, support proofs and exercises, and support learning in mathematics and computing.
In mathematics, a relation between the elements of two sets, formally a set of ordered pairs, which may hold or not for each pair, including relations on a single set with properties such as reflexivity, symmetry, transitivity and antisymmetry, and special kinds such as equivalence relations, partial and total orders, functions and graphs; binary relations are fundamental in set theory, logic, algebra, databases and computer science.
What it is for: Sets of ordered pairs between elements.
It can be explain definition and properties; describe kinds; support proofs; support learning.
Distinguishing features
Ordered pairs
Properties
Composition and inverse
Representable as graphs or matrices
What it looks like
Not physical; a mathematical object.
How it is recognised
Set of ordered pairs
Reflexive, symmetric, transitive and other properties
A function is a special relation; an n-ary relation has more places
Related models
is a kind of - category
is related to - relations between points
is related to - relations between units
is related to - order relations on real numbers
In practice
Families and kinds
equivalence relations
partial and total orders
functions as relations
relations in databases
relations as directed graphs
Standards and regulation
Mathematical notation conventions
Curriculum standards for discrete mathematics
No regulation of the concept
Failure modes and hazards
Confusing relation properties
Confusing relations with functions
Notation ambiguity
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.
Understand What a binary relation is.
Mathematics.
Concept
Definition.
Concept
Concept.
- What is a binary relation formally, and how are domain, codomain, composition and inverse defined? definition
- Which entry fits set theory? action
Properties
Properties.
Properties
Properties.
- What are reflexivity, symmetry, antisymmetry, transitivity and totality, and how are they checked? definition
- Which property is meant? boundary
Kinds Important kinds.
Mathematics.
Equivalence
Equivalence and order.
Equivalence
Equivalence.
- How do equivalence relations, partial orders and total orders arise, and what are their uses? definition
- Which entry fits order theory? action
Functions
Functions and graphs.
Functions
Functions.
- How are functions and directed graphs special cases or representations of relations? definition
- Which entry fits graph theory? action
Apply Applications and proofs.
Practice.
Prove
Proving properties.
Prove
Proof.
- How can it be proved that this relation has or lacks a given property? action
- Which entry fits proof techniques? action
Computing
Computing.
Computing
Computing.
- How are relations used in databases, logic programming and formal methods? provenance
- Which entry fits relational databases? action
Learn History and teaching.
Education.
History
History.
History
History.
- How did the theory of relations develop from De Morgan and Peirce to modern set theory? provenance
- Which references are standard? provenance
Teach
Teaching.
Teach
Teaching.
- How can relations be taught in discrete mathematics? action
- Which misconceptions arise? provenance
What the second pass must settle
- Should equivalence relations be a separate entry?
- How should textbooks be linked?
- How should exercises be linked?