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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 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.
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 .
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, 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, 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.
A relation is transitive if it is closed under this operation, and the transitive closure of a relation is its closure under this operation. A preorder is a relation that is reflective and transitive. It follows that the reflexive transitive closure of a relation is the smallest preorder containing it