Safe Haskell	Safe
Language	Haskell2010

Test.Property.Relation.Reflexive

Description

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

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

For example, the relation $y = x^2$ is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pairs $(0, 0)$ and $(2, 4)$ , but not $(2, 2)$ .

The latter two facts also rule out quasi-reflexivity.

Synopsis

reflexive :: (r -> r -> Bool) -> r -> Bool
irreflexive :: (r -> r -> Bool) -> r -> Bool
coreflexive :: Eq r => (r -> r -> Bool) -> r -> r -> Bool
coreflexive_on :: (r -> r -> Bool) -> (r -> r -> Bool) -> r -> r -> Bool
quasireflexive :: (r -> r -> Bool) -> r -> r -> Bool

Documentation

reflexive :: (r -> r -> Bool) -> r -> Bool Source #

$\forall a: (a \# a)$

For example, ≥ is a reflexive relation but > is not.

irreflexive :: (r -> r -> Bool) -> r -> Bool Source #

$\forall a: \neg (a \# a)$

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

coreflexive :: Eq r => (r -> r -> Bool) -> r -> r -> Bool Source #

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

For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. The equality relation is the only example of a relation that is both reflexive and coreflexive, and any coreflexive relation is a subset of the equality relation.

coreflexive_on :: (r -> r -> Bool) -> (r -> r -> Bool) -> r -> r -> Bool Source #

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

quasireflexive :: (r -> r -> Bool) -> r -> r -> Bool Source #

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

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