Lévy's modulus of continuity theorem

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. Informally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere.

Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

$(a\mathrm{T}b)\wedge \mathrm{\neg }(b\mathrm{T}a)\wedge (b\mathrm{T}c)\wedge \mathrm{\neg }(c\mathrm{T}b)\Rightarrow (a\mathrm{T}c)\wedge \mathrm{\neg }(c\mathrm{T}a).$

If the relation is also antisymmetric, T is transitive.

Alternately, for a relation T, define the asymmetric part P:

$(a\mathrm{P}b)\iff (a\mathrm{T}b)\wedge \mathrm{\neg }(b\mathrm{T}a).$

Then T is quasitransitive iff P is transitive.

Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic example is a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but who prefers 12 grams of sugar to 10.

See also

• Intransitivity
• Reflexive relation

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