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The relation "is the birth parent of" on a set of people is not a transitive relation. However, in biology the need often arises to consider birth parenthood over an arbitrary number of generations: the relation "is a birth ancestor of" is a transitive relation and it is the transitive closure of the relation "is the birth parent of".
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
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). The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere.
[8] [9] This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality. [10] Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial ...
The composition of relations R ∘ R is the relation S defined by setting xSz to be true for a pair of elements x and z in X whenever there exists y in X with xRy and yRz both true. R is idempotent if R = S. Equivalently, relation R is idempotent if and only if the following two properties are true: R is a transitive relation, meaning that R ...
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.
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.
A binary relation that is antisymmetric, transitive, and reflexive (but not necessarily total) is a partial order. A group with a compatible total order is a totally ordered group. There are only a few nontrivial structures that are (interdefinable as) reducts of a total order. Forgetting the orientation results in a betweenness relation.