Quasitransitive relation 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 .A binary relation T over a set X is quasitransitive if for all a , b , and c in X the following holds:antisymmetric , T is transitive.define the asymmetric or "strict" part P: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.
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