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	DonaldsonFreedmanExoticR4 Donaldson Freedman exotic R4 DonaldsonsTheorem Donaldson’s theorem DoobsInequalities DoobsOptionalSamplingTheorem Doob’s inequalities Doob’s optional sampling theorem DoorSpace door space DorothyBernstein Dorothy Bernstein DotProduct dot product DouadyRabbit Douady rabbit DoubleAngleIdentity double angle identity DoubleCoset double coset DoubleFactorial double factorial DoubleGroupoidWithConnection double groupoid with connection DoubleMersenneNumber