symmetric relation

A relation $\\mathcal{R}$ on a set $A$ is symmetric if and only ifwhenever $x\\mathcal{R}y$ for some $x,y\\in A$ then also $y\\mathcal{R}x$ .

An example of a symmetric relation on $\\{a,b,c\\}$ is $\\{(a,a),(c,b),(b,c),(a,c),(c,a)\\}$ .One relation that is not symmetric is $\\mathcal{R}=\\{(b,b),(a,b),(b,a),(c,b)\\}$ ,because $(c,b)\\in\\mathcal{R}$ but $(b,c)\otin\\mathcal{R}$ .

On a finite set with $n$ elements there are $2^{n^{2}}$ relations,of which $2^{\\frac{n^{2}+n}{2}}$ are symmetric.

A relation $\\mathcal{R}$ that is both symmetric and antisymmetrichas the property that $x\\mathcal{R}y$ implies $x=y$ .On a finite set with $n$ elements there are only $2^{n}$ such relations.

单词	SymmetricRelation
释义	symmetric relation A relation $\\mathcal{R}$ on a set $A$ is symmetric if and only ifwhenever $x\\mathcal{R}y$ for some $x,y\\in A$ then also $y\\mathcal{R}x$ . An example of a symmetric relation on $\\{a,b,c\\}$ is $\\{(a,a),(c,b),(b,c),(a,c),(c,a)\\}$ .One relation that is not symmetric is $\\mathcal{R}=\\{(b,b),(a,b),(b,a),(c,b)\\}$ ,because $(c,b)\\in\\mathcal{R}$ but $(b,c)\otin\\mathcal{R}$ . On a finite set with $n$ elements there are $2^{n^{2}}$ relations,of which $2^{\\frac{n^{2}+n}{2}}$ are symmetric. A relation $\\mathcal{R}$ that is both symmetric and antisymmetrichas the property that $x\\mathcal{R}y$ implies $x=y$ .On a finite set with $n$ elements there are only $2^{n}$ such relations.
随便看	52UniquenessOfInductiveTypes 5.2 Uniqueness of inductive types 53Wtypes 5.3 W-types 54InductiveTypesAreInitialAlgebras 5.4 Inductive types are initial algebras 55HomotopyinductiveTypes 5.5 Homotopy-inductive types 56TheGeneralSyntaxOfInductiveDefinitions 5.6 The general syntax of inductive definitions 57GeneralizationsOfInductiveTypes 5.7 Generalizations of inductive types 58IdentityTypesAndIdentitySystems 5.8 Identity types and identity systems 5Entanglement 5. Entanglement 5Induction 5. Induction 610Quotients 6.10 Quotients 611Algebra 6.11 Algebra 612TheFlatteningLemma 6.12 The flattening lemma 613TheGeneralSyntaxOfHigherInductiveDefinitions