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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".
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 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
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure does not relate a woman with her paternal grandmother. On the other ...
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 transitive closure of a DAG is the graph with the most edges that has the same reachability relation as the DAG. It has an edge u → v for every pair of vertices (u, v) in the reachability relation ≤ of the DAG, and may therefore be thought of as a direct translation of the reachability relation ≤ into graph-theoretic
Transitive for all x, y, z ∈ X, if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric. [10] For example, "is ancestor of" is a transitive relation, while "is parent of" is not. Antitransitive for all x, y, z ∈ X, if xRy and yRz then never xRz. Co-transitive if the complement of R is transitive.
One of them is the transitive closure of a binary relation. Given a domain D, let binary relation R be a subset of D×D. The transitive closure R + of R is the smallest subset of D×D that contains R and satisfies the following condition: