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The relation "≥" between real numbers is reflexive and transitive, but not symmetric. For example, 7 ≥ 5 but not 5 ≥ 7. The relation "has a common factor greater than 1 with" between natural numbers greater than 1, is reflexive and symmetric, but not transitive. For example, the natural numbers 2 and 6 have a common factor greater than 1 ...
For example, the Spanish reflexive construct "se hundió el barco" ("the boat sank") has no reflexive equivalent in some Slavic languages (which use an intransitive equivalent of sink), though for example Czech and Slovak do use a reflexive verb: "loď se potopila"/"loď sa potopila". Reflexive verbs can have a variety of uses and meanings ...
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations.
A transitive relation is asymmetric if and only if it is irreflexive. [6] A transitive relation need not be reflexive. When it is, it is called a preorder. For example, on set X = {1,2,3}: R = { (1,1), (2,2), (3,3), (1,3), (3,2) } is reflexive, but not transitive, as the pair (1,2) is absent,
The resulting relation is reflexive since the preorder is reflexive; transitive by applying the transitivity of twice; and symmetric by definition. Using this relation, it is possible to construct a partial order on the quotient set of the equivalence, S / ∼ , {\displaystyle S/\sim ,} which is the set of all equivalence classes of ∼ ...
A reflexive, weak, [1] or non-strict partial order, [2] commonly referred to simply as a partial order, is a homogeneous relation ≤ on a set that is reflexive, antisymmetric, and transitive. That is, for all a , b , c ∈ P , {\displaystyle a,b,c\in P,} it must satisfy:
An example is the relation being a sibling of: no one is their own sibling. [73] These structural features are used to define further types of relations, like equivalence and strict partial order. An equivalence relation is a relation that is reflexive, symmetric, and transitive, like equality expressed through the symbol "=". [74]
The Euclidean property along with reflexivity yields symmetry and transitivity. (The Euclidean property can be obtained, as well, from symmetry and transitivity.) Hence if the accessibility relation R is reflexive and Euclidean, R is provably symmetric and transitive as well. Hence for models of S5, R is an equivalence relation, because R is ...