Relations
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3, and likewise between 3 and 4, but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false. As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not.
| Reflexive | ∀ a ∈ X: aRa |
| Symmetric | ∀ a, b ∈ X: (aRb ⇒ bRa) |
| Antisymmetric | If aRb with a ≠ b then bRa must not hold |
| Transitive | For all a, b, c ∈ X, if aRb and bRc, then aRc |
Result
dom(R) =
im(R) =
R-1 = {}
reflexive:
symmetric:
antisymmetric:
transitive:
Partition of equivalent classes: {}