Relations

Relation axioms

• null (empty) relation

• full (total) relation

• reflexivity

  • reflexive

  • non-reflexive

  • Irreflexive

  • non-irreflexive

  • coreflexive

• Symmetry

  • symmetric

  • antisymmetry

  • asymmetry

• Transitivity

  • transitive

Relation axioms

• Reflexivity

  • Reflexivity: reflexive relation if a is related to itself, aRa

  • Irreflexivity

  • Coreflexivity

• Symmetry

  • Symmetry

  • Antisymmetry

  • Asymmetry

• Transitivity

  • Transitivity

• Reflexivity

  • reflefive, Re: Id+

  • non-reflefive, !Re

  • irreflefive, iR

  • non-irreflefive, !iR

  • coreflexive, cR

  • non-coreflexive, !cR

• Symmerty

  • symmertic, Sy

  • non-symmertic, !Sy

  • anti-symmertic, vS

  • non-antisymmertic, !vS

  • asymmertic, aS

  • non-asymmertic, !aS

• Transitivity

  • transitive, Tr

  • non-transitive, !Tr

• reflexive: Sy+Tr+Serial

• equivalence, EQ = Re+Sy+Tr

• partial equivalence, pEQ: Sy+Tr

• partial order: pOrd = Re+vS+Tr

• linear (total) order: partial order that is total, Re+vS+Tr+

• linear (total) order: partial order that is total, Re+vS+Tr+Total

• well-order: linear order where every non-empty subset has a least element.

• The relationship of one set being a subset of another is called inclusion or sometimes containment.