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The transitive extension of R 1 would be denoted by R 2, and continuing in this way, in general, the transitive extension of R i would be R i + 1. The transitive closure of R, denoted by R* or R ∞ is the set union of R, R 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]
Transitivity is a linguistics property that relates to whether a verb, participle, or gerund denotes a transitive object. It is closely related to valency , which considers other arguments in addition to transitive objects.
Reflexive and transitive: The relation ≤ on N. Or any preorder; Symmetric and transitive: The relation R on N, defined as aRb ↔ ab ≠ 0. Or any partial equivalence relation; Reflexive and symmetric: The relation R on Z, defined as aRb ↔ "a − b is divisible by at least one of 2 or 3." Or any dependency relation.
The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation. An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y".
By definition, every strict weak order is a strict partial order. The set of subsets of a given set (its power set) ordered by inclusion (see Fig. 1). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring. The set of natural numbers equipped with the relation of divisibility. (see Fig. 3 and Fig. 6)
Vertex-transitive graph, a graph whose automorphism group acts transitively upon its vertices; Transitive set a set A such that whenever x ∈ A, and y ∈ x, then y ∈ A; Topological transitivity property of a continuous map for which every open subset U' of the phase space intersects every other open subset V, when going along trajectory
A transitive relation is asymmetric if and only if it is irreflexive: [4] if and , transitivity gives , contradicting irreflexivity. Such a relation is a strict partial order . R {\displaystyle R} is irreflexive and satisfies semiorder property 1 (there do not exist two mutually incomparable two-point linear orders)
The action of G on X is called transitive if for any two points x, y ∈ X there exists a g ∈ G so that g ⋅ x = y. The action is simply transitive (or sharply transitive, or regular) if it is both transitive and free. This means that given x, y ∈ X the element g in the definition of transitivity is unique.