Symmetric closure

In mathematics, the symmetric closure of a binary relation $R$ on a set $X$ is the smallest symmetric relation on $X$ that contains $R.$

For example, if $X$ is a set of airports and $xRy$ means "there is a direct flight from airport $x$ to airport $y$ ", then the symmetric closure of $R$ is the relation "there is a direct flight either from $x$ to $y$ or from $y$ to $x$ ". Or, if $X$ is the set of humans and $R$ is the relation 'parent of', then the symmetric closure of $R$ is the relation " $x$ is a parent or a child of $y$ ".

Definition

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The symmetric closure $S$ of a relation $R$ on a set $X$ is given by $S=R\cup \{(y,x):(x,y)\in R\}.$

In other words, the symmetric closure of $R$ is the union of $R$ with its converse relation, $R^{\operatorname {T} }.$

References

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Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

v t e Order theory
Topics Glossary Category
Key concepts	Binary relation Boolean algebra Cyclic order Lattice Partial order Preorder Total order Weak ordering
Results	Boolean prime ideal theorem Cantor–Bernstein theorem Cantor's isomorphism theorem Dilworth's theorem Dushnik–Miller theorem Hausdorff maximal principle Knaster–Tarski theorem Kruskal's tree theorem Laver's theorem Mirsky's theorem Szpilrajn extension theorem Zorn's lemma
Properties & Types (list)	Antisymmetric Asymmetric Boolean algebra topics Completeness Connected Covering Dense Directed (Partial) Equivalence Foundational Heyting algebra Homogeneous Idempotent Lattice Bounded Complemented Complete Distributive Join and meet Reflexive Partial order Chain-complete Graded Eulerian Strict Prefix order Preorder Total Semilattice Semiorder Symmetric Total Tolerance Transitive Well-founded Well-quasi-ordering (Better) (Pre) Well-order
Constructions	Composition Converse/Transpose Lexicographic order Linear extension Product order Reflexive closure Series-parallel partial order Star product Symmetric closure Transitive closure
Topology & Orders	Alexandrov topology & Specialization preorder Ordered topological vector space Normal cone Order topology Order topology Topological vector lattice Banach Fréchet Locally convex Normed
Related	Antichain Cofinal Cofinality Comparability Graph Duality Filter Hasse diagram Ideal Net Subnet Order morphism Embedding Isomorphism Order type Ordered field Positive cone of an ordered field Ordered vector space Partially ordered Positive cone of an ordered vector space Riesz space Partially ordered group Positive cone of a partially ordered group Upper set Young's lattice

Definition

See also

References

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