Quasitransitive relation
The mathematical notion of quasitransitivity is a weakened version of transitivity that is used in social [choice theory] and microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept was introduced by to study the consequences of Arrow's theorem.
Formal definition
A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:If the relation is also antisymmetric, T is transitive.
Alternately, for a relation T, define the asymmetric or "strict" part P:
Then T is quasitransitive if and only if P is transitive.
Examples
s are assumed to be quasitransitive in some economic contexts. The classic example is a person indifferent between 7 and 8 grams of sugar and indifferent between 8 and 9 grams of sugar, but who prefers 9 grams of sugar to 7. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity of certain relations to quasitransitivity.Properties
- A relation R is quasitransitive if, and only if, it is the disjoint union of a symmetric relation J and a transitive relation P. J and P are not uniquely determined by a given R; however, the P from the only-if part is minimal.
- As a consequence, each symmetric relation is quasitransitive, and so is each transitive relation. Moreover, an antisymmetric and quasitransitive relation is always transitive.
- The relation from the above sugar example,, is quasitransitive, but not transitive.
- A quasitransitive relation needn't be acyclic: for every non-empty set A, the universal relation A×A is both cyclic and quasitransitive.
- A relation is quasitransitive if, and only if, its complement is.
- Similarly, a relation is quasitransitive if, and only if, its converse is.