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An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y". The transitive closure of this relation is "some day x comes after a day y on the calendar", which is trivially true for all days of the week x and y (and thus equivalent to the Cartesian square , which is " x and y are both ...
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 1, R 2, ... . [8] The transitive closure of a relation is a transitive relation. [8]
The transitive closure of a set is the smallest (with respect to inclusion) transitive set that includes (i.e. ()). [2] Suppose one is given a set X {\displaystyle X} , then the transitive closure of X {\displaystyle X} is
For example, the natural numbers ... Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation ...
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 ...
The Floyd–Warshall algorithm is an example of dynamic programming, and was published in its currently recognized form by Robert Floyd in 1962. [3] However, it is essentially the same as algorithms previously published by Bernard Roy in 1959 [4] and also by Stephen Warshall in 1962 [5] for finding the transitive closure of a graph, [6] and is closely related to Kleene's algorithm (published ...
The RS extension is the transitive closure of these relations. Pairwise dominance (PD) The PD ... So, for example, a responsive order can have both {} ...
The two preceding examples are power sets, which are Boolean algebras under the usual set theoretic operations of union, intersection, and complement. This justifies calling them Boolean action algebras. The relational example constitutes a relation algebra equipped with an operation of reflexive transitive closure. Note that every Boolean ...