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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.
If a relation is transitive then its transitive extension is itself, that is, if R is a transitive relation then R 1 = R. 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 ...
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".
[1]: 322 Conversely, British English favours fitted as the past tense of fit generally, whereas the preference of American English is more complex: AmE prefers fitted for the metaphorical sense of having made an object [adjective-]"fit" (i.e., suited) for a purpose; in spatial transitive contexts, AmE uses fitted for the sense of having made an ...
All definitions tacitly require the homogeneous relation be transitive: for all ,,, if and then . A term's definition may require additional properties that are not listed in this table. Fig. 1 The Hasse diagram of the set of all subsets of a three-element set { x , y , z } , {\displaystyle \{x,y,z\},} ordered by inclusion .
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
In order to have transitive preferences, a person, player, or agent that prefers choice option A to B and B to C must prefer A to C. The most discussed logical property of preferences are the following: A≽B ∧ B≽C → A≽C (transitivity of weak preference) A~B ∧ B~C → A~C (transitivity of indifference)
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.