Transitive/intransitive relation Glossary
Definition
A binary relation is transitive (on a domain of discourse) iff whenever it relates one thing to another and that second thing to a third, it also relates the first thing to the third. That is, a transitive relation R satisfies the condition
ALLxALLy(Rxy IMP ALLz(Ryz IMP Rxz))
R is intransitive iff whenever it relates one thing to another and the second to a third, it does not relate the first to the third. That is, iff it satisfies
ALLxALLy(Rxy IMP NOTSOMEz(Rxz AND Ryz))
Comments
Note that 'intransitive' does not mean 'not transitive'. It is a much stronger condition. Most relations, in fact, are neither transitive nor intransitive.
Where R is a relation between sets of things and single things, as in the case of the relation of logical consequence, it may satisfy a more general condition also called transitivity (or extended transitivity), meaning that R(s,a) holds whenever R(s,b) holds for every object b in a set t, and R(t,a) holds.
Examples
- The relation of order "less than" in the standard number systems, written '<', is transitive.
- The adjacency relation between objects in a line (relating a and b iff a is next to b) is intransitive.