Test.Relation.Symmetric

See https://en.wikipedia.org/wiki/Binary_relation#Properties.

Note that these properties do not exhaust all of the possibilities.

As an example over the natural numbers, the relation \(a \# b \) defined by \( a > 2 \) is neither symmetric nor antisymmetric, let alone asymmetric.

Documentation

symmetric :: (r -> r -> Bool) -> r -> r -> Bool

\( \forall a, b: (a \# b) \Leftrightarrow (b \# a) \)

For example, "is a blood relative of" is a symmetric relation, because A is a blood relative of B if and only if B is a blood relative of A.

asymmetric :: (r -> r -> Bool) -> r -> r -> Bool

\( \forall a, b: (a \# b) \Rightarrow \neg (b \# a) \)

For example, > is an asymmetric relation, but ≥ is not.

A relation is asymmetric if and only if it is both antisymmetric and irreflexive.

antisymmetric :: Eq r => (r -> r -> Bool) -> r -> r -> Bool

\( \forall a, b: (a \# b) \wedge (b \# a) \Rightarrow a \equiv b \)

For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).

antisymmetric_on :: (r -> r -> Bool) -> (r -> r -> Bool) -> r -> r -> Bool

\( \forall a, b: (a \# b) \wedge (b \# a) \Rightarrow a \doteq b \)