predicates {relations}R Documentation

Relation Predicates

Description

Predicate functions for testing for binary relations and endorelations, and special kinds thereof.

Usage

relation_is_antisymmetric(x)
relation_is_asymmetric(x)
relation_is_bijective(x)
relation_is_binary(x)
relation_is_complete(x)
relation_is_coreflexive(x)
relation_is_endorelation(x)
relation_is_equivalence(x)
relation_is_functional(x)
relation_is_injective(x)
relation_is_irreflexive(x)
relation_is_left_total(x)
relation_is_linear_order(x)
relation_is_partial_order(x)
relation_is_reflexive(x)
relation_is_right_total(x)
relation_is_strict_linear_order(x)
relation_is_strict_partial_order(x)
relation_is_surjective(x)
relation_is_symmetric(x)
relation_is_tournament(x)
relation_is_transitive(x)
relation_is_weak_order(x)
relation_is_preference(x)
relation_is_preorder(x)
relation_is_quasiorder(x)

Arguments

x an object inheriting from class relation.

Details

A binary relation is a relation with arity 2. An endorelation R on a set X is a relation with domain D(R) = (X, X), i.e., a binary relation on X.

Let us write x R y iff (x, y) is contained in R.

A binary relation R is called

left-total:
for all x there is at least one y such that x R y.
right-total:
for all y there is at least one x such that x R y.
functional:
for all x there is at most one y such that x R y.
surjective:
the same as right-total.
injective:
for all y there is at most one x such that x R y.
bijective:
left-total, right-total, functional and injective.

An endorelation R is called

reflexive:
x R x for all x.
irreflexive:
there is no x such that x R x.
coreflexive:
x R y implies x = y.
symmetric:
x R y implies y R x.
asymmetric:
x R y implies that not y R x.
antisymmetric:
x R y and y R x imply that x = y.
transitive:
x R y and y R z imply that x R z.
complete:
for all x and y, x R y or y R x.

Some combinations of these basic properties have special names because of their widespread use:

preorder:
reflexive and transitive.
quasiorder:
the same as preorder.
equivalence:
a symmetric preorder.
weak order:
complete and transitive.
preference:
the same as weak order.
partial order:
an antisymmetric preorder.
strict partial order:
irreflexive, transitive and antisymmetric.
linear order:
a complete partial order.
strict linear order:
a complete strict partial order.
tournament:
complete and antisymmetric.

If R is a weak order (“weak preference relation”), I = I(R) defined by x I y iff x R y and y R x is an equivalence, the indifference relation corresponding to R.

There seem to be no commonly agreed definitions for order relations: e.g., Fishburn (1972) requires these to be irreflexive.

References

P. C. Fishburn (1972), Mathematics of decision theory. Methods and Models in the Social Sciences 3. Mouton: The Hague.

H. R. Varian (2002), Intermediate Microeconomics: A Modern Approach. 6th Edition. W. W. Norton & Company.


[Package relations version 0.2-0 Index]