Symmetric / asymmetric relation Glossary
Definition
A binary relation is symmetric (on a domain of discourse) iff whenever it relates two things in one direction, it relates them in the other direction as well. That is, a symmetric relation R satisfies the condition
ALLxALLy(Rxy IMP Ryx)
R is asymmetric iff it only ever relates two things in one direction. That is, iff it satisfies
ALLxALLy(Rxy IMP NOTRyx)
R is antisymmetric iff it never relates two distinct things to each other (in both directions). That is, iff it satisfies
ALLxALLy((Rxy AND Ryx) IMP x=y)
Comments
Note that 'asymmetric' does not mean 'not symmetric'. Nor does 'antisymmetric'. They are both much stronger conditions.
Examples
- The spouse relation in the domain of people is symmetric: if a is married to b, then b is married to a.
- The relation of order "less than" in the standard number systems, written '<', is asymmetric.
- The relation of inclusion between sets, written '⊆', is antisymmetric but not asymmetric.