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In mathematics, a binary relation R on a set X is transitive if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c. Every partial order and every equivalence relation is transitive.
A transitive relation is irreflexive if and only if it is asymmetric. [13] For example, "is ancestor of" is a transitive relation, while "is parent of" is not. Connected for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, on the natural numbers, < is connected, while "is a divisor of " is not (e.g. neither 5R7 nor 7R5). Strongly connected
If X has cardinality n, the action of the alternating group is (n − 2)-transitive but not (n − 1)-transitive. The action of the general linear group of a vector space V on the set V ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of v is at least 2).
An irreflexive, strong, [1] or strict partial order is a homogeneous relation < on a set that is irreflexive, asymmetric and transitive; that is, it satisfies the following conditions for all ,,: Irreflexivity : ¬ ( a < a ) {\displaystyle \neg \left(a<a\right)} , i.e. no element is related to itself (also called anti-reflexive).
In mathematics, the transitive closure R + of a homogeneous binary relation R on a set X is the smallest relation on X that contains R and is transitive.For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets R + is the unique minimal transitive superset of R.
In mathematics, a binary relation associates elements of one set called the domain with elements of ... the smallest transitive relation over containing , ...
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Total relation. Synonym for Connected relation. Transitive relation. A relation R on a set X is transitive, if x R y and y R z imply x R z, for all elements x, y, z in X. Transitive closure. The transitive closure R ∗ of a relation R consists of all pairs x,y for which there cists a finite chain x R a, a R b, ..., z R y. [1]