symmetric relation
Let an agent define symmetric relations, give examples and counterexamples, explain related properties and the symmetric closure, and separate the mathematical concept from unrelated aliases in the registry entry.
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 define symmetric relations, give examples and counterexamples, explain related properties and the symmetric closure, and separate the mathematical concept from unrelated aliases in the registry entry.
A binary relation on a set in which whenever a is related to b, b is related to a, such as equality, being a sibling of, or being connected to; symmetric relations include equivalence relations and compatibility relations, contrast with antisymmetric and asymmetric relations such as ordering, and any relation can be extended to its symmetric closure by adding the reverse pairs.
What it is for: A basic property of relations in mathematics and logic.
It can be define and test symmetry; give examples and counterexamples; explain related properties and closure; separate unrelated aliases.
Distinguishing features
Symmetry property
Undirected representation
Part of equivalence relations
Symmetric closure
What it looks like
Not a visible object; an undirected graph or a symmetric matrix when drawn.
How it is recognised
If a R b then b R a
Undirected graph or symmetric adjacency matrix
Antisymmetric relations such as less than or equal are different; reflexive and transitive are separate properties
Related models
is a kind of - in registry terms, a relation on one set
is a kind of - with the symmetry property
is part of - together with reflexivity and transitivity
is contrasted with - the opposite property
In practice
Families and kinds
equivalence relations
compatibility and comparability relations
the empty relation, symmetric vacuously
symmetric closures of relations
symmetric relations in graphs and matrices
relations in specific domains such as transversality in geometry or genetic relationship, listed as aliases
Standards and regulation
ISO 80000-2 for mathematical notation
Failure modes and hazards
Confusing symmetric with reflexive or transitive
Assuming symmetry in directed data
Sense confusion with unrelated aliases
Also called
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.
Define What a symmetric relation is.
Definition.
Definition
Definition and examples.
Definition
Definition.
- What is a symmetric relation, and what are examples and counterexamples? definition
- Is the relation symmetric, antisymmetric, asymmetric or none? boundary
Properties
Related properties.
Properties
Properties.
- How does symmetry combine with reflexivity and transitivity in equivalence and compatibility relations? definition
- Which entry fits equivalence relation? action
Compute Testing and closure.
Computation.
Test
Testing symmetry.
Test
Test.
- How is symmetry tested for a given relation, matrix or graph? action
- Is the relation in question symmetric? measurement
Closure
Symmetric closure.
Closure
Closure.
- How is the symmetric closure constructed, and what are its properties? action
- Which entry fits closure of a relation? action
Represent Representations.
Theory.
Graphs
Graphs and matrices.
Graphs
Graphs.
- How do symmetric relations correspond to undirected graphs and symmetric matrices? provenance
- Which entry fits undirected graph? action
Counting
Counting relations.
Counting
Counting.
- How many symmetric relations exist on a finite set? provenance
- Which references are standard? provenance
Context Uses and teaching.
Context.
Uses
Uses.
Uses
Uses.
- Where do symmetric relations appear in mathematics, computing and the sciences? provenance
- Which entry fits the specific application? action
Teach
Teaching.
Teach
Teaching.
- How are relation properties taught, and which are confused? provenance
- Which entry fits discrete mathematics? action
What the second pass must settle
- Should symmetric closure be a separate entry?
- How should logic references be linked?
- The registry entry has merged aliases for unrelated concepts such as arene substitution patterns; should they be removed?